Generalized Hermite polynomials for Painlevé IV

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Polynomials

$m$, $n$

$\widehat H_{m,n}(x)$

1, 1:

1, 2:

x^2 + 1/2

2, 1:

x^2 - 1/2

1, 3:

x^3 + 3/2*x

3, 1:

x^3 - 3/2*x

1, 4:

x^4 + 3*x^2 + 3/4

2, 2:

x^4 + 3/4

4, 1:

x^4 - 3*x^2 + 3/4

1, 5:

x^5 + 5*x^3 + 15/4*x

5, 1:

x^5 - 5*x^3 + 15/4*x

1, 6:

x^6 + 15/2*x^4 + 45/4*x^2 + 15/8

2, 3:

x^6 + 3/2*x^4 + 9/4*x^2 - 9/8

3, 2:

x^6 - 3/2*x^4 + 9/4*x^2 + 9/8

6, 1:

x^6 - 15/2*x^4 + 45/4*x^2 - 15/8

1, 7:

x^7 + 21/2*x^5 + 105/4*x^3 + 105/8*x

7, 1:

x^7 - 21/2*x^5 + 105/4*x^3 - 105/8*x

1, 8:

x^8 + 14*x^6 + 105/2*x^4 + 105/2*x^2 + 105/16

2, 4:

x^8 + 4*x^6 + 15/2*x^4 + 45/16

4, 2:

x^8 - 4*x^6 + 15/2*x^4 + 45/16

8, 1:

x^8 - 14*x^6 + 105/2*x^4 - 105/2*x^2 + 105/16

1, 9:

x^9 + 18*x^7 + 189/2*x^5 + 315/2*x^3 + 945/16*x

3, 3:

x^9 + 9/2*x^5 - 135/16*x

9, 1:

x^9 - 18*x^7 + 189/2*x^5 - 315/2*x^3 + 945/16*x

1, 10:

x^10 + 45/2*x^8 + 315/2*x^6 + 1575/4*x^4 + 4725/16*x^2 + 945/32

2, 5:

x^10 + 15/2*x^8 + 45/2*x^6 + 75/4*x^4 + 225/16*x^2 - 225/32

5, 2:

x^10 - 15/2*x^8 + 45/2*x^6 - 75/4*x^4 + 225/16*x^2 + 225/32

10, 1:

x^10 - 45/2*x^8 + 315/2*x^6 - 1575/4*x^4 + 4725/16*x^2 - 945/32

1, 11:

x^11 + 55/2*x^9 + 495/2*x^7 + 3465/4*x^5 + 17325/16*x^3 + 10395/32*x

11, 1:

x^11 - 55/2*x^9 + 495/2*x^7 - 3465/4*x^5 + 17325/16*x^3 - 10395/32*x

1, 12:

x^12 + 33*x^10 + 1485/4*x^8 + 3465/2*x^6 + 51975/16*x^4 + 31185/16*x^2 + 10395/64

2, 6:

x^12 + 12*x^10 + 225/4*x^8 + 105*x^6 + 1575/16*x^4 + 1575/64

3, 4:

x^12 + 3*x^10 + 45/4*x^8 + 15/2*x^6 - 225/16*x^4 + 675/16*x^2 + 675/64

4, 3:

x^12 - 3*x^10 + 45/4*x^8 - 15/2*x^6 - 225/16*x^4 - 675/16*x^2 + 675/64

6, 2:

x^12 - 12*x^10 + 225/4*x^8 - 105*x^6 + 1575/16*x^4 + 1575/64

12, 1:

x^12 - 33*x^10 + 1485/4*x^8 - 3465/2*x^6 + 51975/16*x^4 - 31185/16*x^2 + 10395/64

1, 13:

x^13 + 39*x^11 + 2145/4*x^9 + 6435/2*x^7 + 135135/16*x^5 + 135135/16*x^3 + 135135/64*x

13, 1:

x^13 - 39*x^11 + 2145/4*x^9 - 6435/2*x^7 + 135135/16*x^5 - 135135/16*x^3 + 135135/64*x

1, 14:

x^14 + 91/2*x^12 + 3003/4*x^10 + 45045/8*x^8 + 315315/16*x^6 + 945945/32*x^4 + 945945/64*x^2 + 135135/128

2, 7:

x^14 + 35/2*x^12 + 483/4*x^10 + 3045/8*x^8 + 9555/16*x^6 + 11025/32*x^4 + 11025/64*x^2 - 11025/128

7, 2:

x^14 - 35/2*x^12 + 483/4*x^10 - 3045/8*x^8 + 9555/16*x^6 - 11025/32*x^4 + 11025/64*x^2 + 11025/128

14, 1:

x^14 - 91/2*x^12 + 3003/4*x^10 - 45045/8*x^8 + 315315/16*x^6 - 945945/32*x^4 + 945945/64*x^2 - 135135/128

1, 15:

x^15 + 105/2*x^13 + 4095/4*x^11 + 75075/8*x^9 + 675675/16*x^7 + 2837835/32*x^5 + 4729725/64*x^3 + 2027025/128*x

3, 5:

x^15 + 15/2*x^13 + 135/4*x^11 + 525/8*x^9 + 675/16*x^7 + 4725/32*x^5 - 7875/64*x^3 - 23625/128*x

5, 3:

x^15 - 15/2*x^13 + 135/4*x^11 - 525/8*x^9 + 675/16*x^7 - 4725/32*x^5 - 7875/64*x^3 + 23625/128*x

15, 1:

x^15 - 105/2*x^13 + 4095/4*x^11 - 75075/8*x^9 + 675675/16*x^7 - 2837835/32*x^5 + 4729725/64*x^3 - 2027025/128*x

1, 16:

x^16 + 60*x^14 + 1365*x^12 + 15015*x^10 + 675675/8*x^8 + 945945/4*x^6 + 4729725/16*x^4 + 2027025/16*x^2 + 2027025/256

2, 8:

x^16 + 24*x^14 + 231*x^12 + 1092*x^10 + 21735/8*x^8 + 6615/2*x^6 + 33075/16*x^4 + 99225/256

4, 4:

x^16 + 15*x^12 - 225/8*x^8 + 7875/16*x^4 + 23625/256

8, 2:

x^16 - 24*x^14 + 231*x^12 - 1092*x^10 + 21735/8*x^8 - 6615/2*x^6 + 33075/16*x^4 + 99225/256

16, 1:

x^16 - 60*x^14 + 1365*x^12 - 15015*x^10 + 675675/8*x^8 - 945945/4*x^6 + 4729725/16*x^4 - 2027025/16*x^2 + 2027025/256

1, 17:

x^17 + 68*x^15 + 1785*x^13 + 23205*x^11 + 1276275/8*x^9 + 2297295/4*x^7 + 16081065/16*x^5 + 11486475/16*x^3 + 34459425/256*x

17, 1:

x^17 - 68*x^15 + 1785*x^13 - 23205*x^11 + 1276275/8*x^9 - 2297295/4*x^7 + 16081065/16*x^5 - 11486475/16*x^3 + 34459425/256*x

1, 18:

x^18 + 153/2*x^16 + 2295*x^14 + 69615/2*x^12 + 2297295/8*x^10 + 20675655/16*x^8 + 48243195/16*x^6 + 103378275/32*x^4 + 310134825/256*x^2 + 34459425/512

2, 9:

x^18 + 63/2*x^16 + 405*x^14 + 5355/2*x^12 + 78435/8*x^10 + 314685/16*x^8 + 337365/16*x^6 + 297675/32*x^4 + 893025/256*x^2 - 893025/512

3, 6:

x^18 + 27/2*x^16 + 90*x^14 + 315*x^12 + 4725/8*x^10 + 14175/16*x^8 + 496125/256*x^2 + 165375/512

6, 3:

x^18 - 27/2*x^16 + 90*x^14 - 315*x^12 + 4725/8*x^10 - 14175/16*x^8 + 496125/256*x^2 - 165375/512

9, 2:

x^18 - 63/2*x^16 + 405*x^14 - 5355/2*x^12 + 78435/8*x^10 - 314685/16*x^8 + 337365/16*x^6 - 297675/32*x^4 + 893025/256*x^2 + 893025/512

18, 1:

x^18 - 153/2*x^16 + 2295*x^14 - 69615/2*x^12 + 2297295/8*x^10 - 20675655/16*x^8 + 48243195/16*x^6 - 103378275/32*x^4 + 310134825/256*x^2 - 34459425/512

1, 19:

x^19 + 171/2*x^17 + 2907*x^15 + 101745/2*x^13 + 3968055/8*x^11 + 43648605/16*x^9 + 130945815/16*x^7 + 392837445/32*x^5 + 1964187225/256*x^3 + 654729075/512*x

19, 1:

x^19 - 171/2*x^17 + 2907*x^15 - 101745/2*x^13 + 3968055/8*x^11 - 43648605/16*x^9 + 130945815/16*x^7 - 392837445/32*x^5 + 1964187225/256*x^3 - 654729075/512*x

1, 20:

x^20 + 95*x^18 + 14535/4*x^16 + 72675*x^14 + 6613425/8*x^12 + 43648605/8*x^10 + 654729075/32*x^8 + 654729075/16*x^6 + 9820936125/256*x^4 + 3273645375/256*x^2 + 654729075/1024

2, 10:

x^20 + 40*x^18 + 2655/4*x^16 + 5850*x^14 + 237825/8*x^12 + 176715/2*x^10 + 4833675/32*x^8 + 1091475/8*x^6 + 16372125/256*x^4 + 9823275/1024

4, 5:

x^20 + 5*x^18 + 135/4*x^16 + 75*x^14 + 525/8*x^12 + 1575/8*x^10 + 39375/32*x^8 - 55125/16*x^6 - 826875/256*x^4 - 826875/256*x^2 + 826875/1024

5, 4:

x^20 - 5*x^18 + 135/4*x^16 - 75*x^14 + 525/8*x^12 - 1575/8*x^10 + 39375/32*x^8 + 55125/16*x^6 - 826875/256*x^4 + 826875/256*x^2 + 826875/1024

10, 2:

x^20 - 40*x^18 + 2655/4*x^16 - 5850*x^14 + 237825/8*x^12 - 176715/2*x^10 + 4833675/32*x^8 - 1091475/8*x^6 + 16372125/256*x^4 + 9823275/1024

20, 1:

x^20 - 95*x^18 + 14535/4*x^16 - 72675*x^14 + 6613425/8*x^12 - 43648605/8*x^10 + 654729075/32*x^8 - 654729075/16*x^6 + 9820936125/256*x^4 - 3273645375/256*x^2 + 654729075/1024

1, 21:

x^21 + 105*x^19 + 17955/4*x^17 + 101745*x^15 + 10683225/8*x^13 + 83329155/8*x^11 + 1527701175/32*x^9 + 1964187225/16*x^7 + 41247931725/256*x^5 + 22915517625/256*x^3 + 13749310575/1024*x

3, 7:

x^21 + 21*x^19 + 819/4*x^17 + 1113*x^15 + 28665/8*x^13 + 59535/8*x^11 + 253575/32*x^9 + 99225/16*x^7 + 2083725/256*x^5 - 3472875/256*x^3 - 10418625/1024*x

7, 3:

x^21 - 21*x^19 + 819/4*x^17 - 1113*x^15 + 28665/8*x^13 - 59535/8*x^11 + 253575/32*x^9 - 99225/16*x^7 + 2083725/256*x^5 + 3472875/256*x^3 - 10418625/1024*x

21, 1:

x^21 - 105*x^19 + 17955/4*x^17 - 101745*x^15 + 10683225/8*x^13 - 83329155/8*x^11 + 1527701175/32*x^9 - 1964187225/16*x^7 + 41247931725/256*x^5 - 22915517625/256*x^3 + 13749310575/1024*x

1, 22:

x^22 + 231/2*x^20 + 21945/4*x^18 + 1119195/8*x^16 + 16787925/8*x^14 + 305540235/16*x^12 + 3360942585/32*x^10 + 21606059475/64*x^8 + 151242416325/256*x^6 + 252070693875/512*x^4 + 151242416325/1024*x^2 + 13749310575/2048

2, 11:

x^22 + 99/2*x^20 + 4125/4*x^18 + 93555/8*x^16 + 631125/8*x^14 + 5207895/16*x^12 + 26185005/32*x^10 + 77182875/64*x^8 + 252130725/256*x^6 + 180093375/512*x^4 + 108056025/1024*x^2 - 108056025/2048

11, 2:

x^22 - 99/2*x^20 + 4125/4*x^18 - 93555/8*x^16 + 631125/8*x^14 - 5207895/16*x^12 + 26185005/32*x^10 - 77182875/64*x^8 + 252130725/256*x^6 - 180093375/512*x^4 + 108056025/1024*x^2 + 108056025/2048

22, 1:

x^22 - 231/2*x^20 + 21945/4*x^18 - 1119195/8*x^16 + 16787925/8*x^14 - 305540235/16*x^12 + 3360942585/32*x^10 - 21606059475/64*x^8 + 151242416325/256*x^6 - 252070693875/512*x^4 + 151242416325/1024*x^2 - 13749310575/2048

1, 23:

x^23 + 253/2*x^21 + 26565/4*x^19 + 1514205/8*x^17 + 25741485/8*x^15 + 540571185/16*x^13 + 7027425405/32*x^11 + 55215485325/64*x^9 + 496939367925/256*x^7 + 1159525191825/512*x^5 + 1159525191825/1024*x^3 + 316234143225/2048*x

23, 1:

x^23 - 253/2*x^21 + 26565/4*x^19 - 1514205/8*x^17 + 25741485/8*x^15 - 540571185/16*x^13 + 7027425405/32*x^11 - 55215485325/64*x^9 + 496939367925/256*x^7 - 1159525191825/512*x^5 + 1159525191825/1024*x^3 - 316234143225/2048*x

1, 24:

x^24 + 138*x^22 + 15939/2*x^20 + 504735/2*x^18 + 77224455/16*x^16 + 231673365/4*x^14 + 7027425405/16*x^12 + 33129291195/16*x^10 + 1490818103775/256*x^8 + 1159525191825/128*x^6 + 3478575575475/512*x^4 + 948702429675/512*x^2 + 316234143225/4096

2, 12:

x^24 + 60*x^22 + 3069/2*x^20 + 21780*x^18 + 3016035/16*x^16 + 2062665/2*x^14 + 57432375/16*x^12 + 31216185/4*x^10 + 2608781175/256*x^8 + 468242775/64*x^6 + 1404728325/512*x^4 + 1404728325/4096

3, 8:

x^24 + 30*x^22 + 819/2*x^20 + 6405/2*x^18 + 248535/16*x^16 + 195615/4*x^14 + 1554525/16*x^12 + 2004345/16*x^10 + 27683775/256*x^8 - 2083725/128*x^6 + 31255875/512*x^4 + 93767625/512*x^2 + 93767625/4096

4, 6:

x^24 + 12*x^22 + 189/2*x^20 + 420*x^18 + 17955/16*x^16 + 4725/2*x^14 + 77175/16*x^12 - 33075/4*x^10 + 1488375/256*x^8 + 3472875/64*x^6 + 52093125/512*x^4 + 52093125/4096

6, 4:

x^24 - 12*x^22 + 189/2*x^20 - 420*x^18 + 17955/16*x^16 - 4725/2*x^14 + 77175/16*x^12 + 33075/4*x^10 + 1488375/256*x^8 - 3472875/64*x^6 + 52093125/512*x^4 + 52093125/4096

8, 3:

x^24 - 30*x^22 + 819/2*x^20 - 6405/2*x^18 + 248535/16*x^16 - 195615/4*x^14 + 1554525/16*x^12 - 2004345/16*x^10 + 27683775/256*x^8 + 2083725/128*x^6 + 31255875/512*x^4 - 93767625/512*x^2 + 93767625/4096

12, 2:

x^24 - 60*x^22 + 3069/2*x^20 - 21780*x^18 + 3016035/16*x^16 - 2062665/2*x^14 + 57432375/16*x^12 - 31216185/4*x^10 + 2608781175/256*x^8 - 468242775/64*x^6 + 1404728325/512*x^4 + 1404728325/4096

24, 1:

x^24 - 138*x^22 + 15939/2*x^20 - 504735/2*x^18 + 77224455/16*x^16 - 231673365/4*x^14 + 7027425405/16*x^12 - 33129291195/16*x^10 + 1490818103775/256*x^8 - 1159525191825/128*x^6 + 3478575575475/512*x^4 - 948702429675/512*x^2 + 316234143225/4096

1, 25:

x^25 + 150*x^23 + 18975/2*x^21 + 664125/2*x^19 + 113565375/16*x^17 + 386122275/4*x^15 + 13514279625/16*x^13 + 75293843625/16*x^11 + 4141161399375/256*x^9 + 4141161399375/128*x^7 + 17392877877375/512*x^5 + 7905853580625/512*x^3 + 7905853580625/4096*x

5, 5:

x^25 + 75/2*x^21 + 1575/16*x^17 + 55125/16*x^13 - 15710625/256*x^9 - 52093125/512*x^5 + 260465625/4096*x

25, 1:

x^25 - 150*x^23 + 18975/2*x^21 - 664125/2*x^19 + 113565375/16*x^17 - 386122275/4*x^15 + 13514279625/16*x^13 - 75293843625/16*x^11 + 4141161399375/256*x^9 - 4141161399375/128*x^7 + 17392877877375/512*x^5 - 7905853580625/512*x^3 + 7905853580625/4096*x

1, 26:

x^26 + 325/2*x^24 + 22425/2*x^22 + 1726725/4*x^20 + 164038875/16*x^18 + 5019589575/32*x^16 + 25097947875/16*x^14 + 326273322375/32*x^12 + 10767019638375/256*x^10 + 53835098191875/512*x^8 + 75369137468625/512*x^6 + 102776096548125/1024*x^4 + 102776096548125/4096*x^2 + 7905853580625/8192

2, 13:

x^26 + 143/2*x^24 + 4407/2*x^22 + 153153/4*x^20 + 6621615/16*x^18 + 92837745/32*x^16 + 213918705/16*x^14 + 1287701415/32*x^12 + 19768764015/256*x^10 + 46088467425/512*x^8 + 30435780375/512*x^6 + 18261468225/1024*x^4 + 18261468225/4096*x^2 - 18261468225/8192

13, 2:

x^26 - 143/2*x^24 + 4407/2*x^22 - 153153/4*x^20 + 6621615/16*x^18 - 92837745/32*x^16 + 213918705/16*x^14 - 1287701415/32*x^12 + 19768764015/256*x^10 - 46088467425/512*x^8 + 30435780375/512*x^6 - 18261468225/1024*x^4 + 18261468225/4096*x^2 + 18261468225/8192

26, 1:

x^26 - 325/2*x^24 + 22425/2*x^22 - 1726725/4*x^20 + 164038875/16*x^18 - 5019589575/32*x^16 + 25097947875/16*x^14 - 326273322375/32*x^12 + 10767019638375/256*x^10 - 53835098191875/512*x^8 + 75369137468625/512*x^6 - 102776096548125/1024*x^4 + 102776096548125/4096*x^2 - 7905853580625/8192

1, 27:

x^27 + 351/2*x^25 + 26325/2*x^23 + 2220075/4*x^21 + 233107875/16*x^19 + 7972289325/32*x^17 + 45176306175/16*x^15 + 677644592625/32*x^13 + 26428139112375/256*x^11 + 161505294575625/512*x^9 + 290709530236125/512*x^7 + 554990921359875/1024*x^5 + 924984868933125/4096*x^3 + 213458046676875/8192*x

3, 9:

x^27 + 81/2*x^25 + 1485/2*x^23 + 31725/4*x^21 + 864675/16*x^19 + 7850115/32*x^17 + 11966535/16*x^15 + 49116375/32*x^13 + 534921975/256*x^11 + 834978375/512*x^9 + 618866325/512*x^7 + 618866325/1024*x^5 - 9282994875/4096*x^3 - 9282994875/8192*x

9, 3:

x^27 - 81/2*x^25 + 1485/2*x^23 - 31725/4*x^21 + 864675/16*x^19 - 7850115/32*x^17 + 11966535/16*x^15 - 49116375/32*x^13 + 534921975/256*x^11 - 834978375/512*x^9 + 618866325/512*x^7 - 618866325/1024*x^5 - 9282994875/4096*x^3 + 9282994875/8192*x

27, 1:

x^27 - 351/2*x^25 + 26325/2*x^23 - 2220075/4*x^21 + 233107875/16*x^19 - 7972289325/32*x^17 + 45176306175/16*x^15 - 677644592625/32*x^13 + 26428139112375/256*x^11 - 161505294575625/512*x^9 + 290709530236125/512*x^7 - 554990921359875/1024*x^5 + 924984868933125/4096*x^3 - 213458046676875/8192*x

1, 28:

x^28 + 189*x^26 + 61425/4*x^24 + 1412775/2*x^22 + 326351025/16*x^20 + 6200669475/16*x^18 + 316234143225/64*x^16 + 677644592625/16*x^14 + 61665657928875/256*x^12 + 226107412405875/256*x^10 + 2034966711652875/1024*x^8 + 1294978816506375/512*x^6 + 6474894082531875/4096*x^4 + 1494206326738125/4096*x^2 + 213458046676875/16384

2, 14:

x^28 + 84*x^26 + 12285/4*x^24 + 64155*x^22 + 13558545/16*x^20 + 14819805/2*x^18 + 2797159365/64*x^16 + 1392566175/8*x^14 + 118238395275/256*x^12 + 50726300625/64*x^10 + 858289006575/1024*x^8 + 127830277575/256*x^6 + 639151387875/4096*x^4 + 273922023375/16384

4, 7:

x^28 + 21*x^26 + 945/4*x^24 + 3255/2*x^22 + 116865/16*x^20 + 368235/16*x^18 + 3380265/64*x^16 + 893025/16*x^14 + 25699275/256*x^12 + 59038875/256*x^10 - 218791125/1024*x^8 - 1093955625/512*x^6 - 5469778125/4096*x^4 - 3281866875/4096*x^2 + 3281866875/16384

7, 4:

x^28 - 21*x^26 + 945/4*x^24 - 3255/2*x^22 + 116865/16*x^20 - 368235/16*x^18 + 3380265/64*x^16 - 893025/16*x^14 + 25699275/256*x^12 - 59038875/256*x^10 - 218791125/1024*x^8 + 1093955625/512*x^6 - 5469778125/4096*x^4 + 3281866875/4096*x^2 + 3281866875/16384

14, 2:

x^28 - 84*x^26 + 12285/4*x^24 - 64155*x^22 + 13558545/16*x^20 - 14819805/2*x^18 + 2797159365/64*x^16 - 1392566175/8*x^14 + 118238395275/256*x^12 - 50726300625/64*x^10 + 858289006575/1024*x^8 - 127830277575/256*x^6 + 639151387875/4096*x^4 + 273922023375/16384

28, 1:

x^28 - 189*x^26 + 61425/4*x^24 - 1412775/2*x^22 + 326351025/16*x^20 - 6200669475/16*x^18 + 316234143225/64*x^16 - 677644592625/16*x^14 + 61665657928875/256*x^12 - 226107412405875/256*x^10 + 2034966711652875/1024*x^8 - 1294978816506375/512*x^6 + 6474894082531875/4096*x^4 - 1494206326738125/4096*x^2 + 213458046676875/16384

1, 29:

x^29 + 203*x^27 + 71253/4*x^25 + 1781325/2*x^23 + 450675225/16*x^21 + 9464179725/16*x^19 + 539458244325/64*x^17 + 1310112879075/16*x^15 + 137561852302875/256*x^13 + 596101359979125/256*x^11 + 6557114959770375/1024*x^9 + 5364912239812125/512*x^7 + 37554385678684875/4096*x^5 + 14443994491801875/4096*x^3 + 6190283353629375/16384*x

29, 1:

x^29 - 203*x^27 + 71253/4*x^25 - 1781325/2*x^23 + 450675225/16*x^21 - 9464179725/16*x^19 + 539458244325/64*x^17 - 1310112879075/16*x^15 + 137561852302875/256*x^13 - 596101359979125/256*x^11 + 6557114959770375/1024*x^9 - 5364912239812125/512*x^7 + 37554385678684875/4096*x^5 - 14443994491801875/4096*x^3 + 6190283353629375/16384*x

1, 30:

x^30 + 435/2*x^28 + 82215/4*x^26 + 8906625/8*x^24 + 614557125/16*x^22 + 28392539175/32*x^20 + 899097073875/64*x^18 + 19651693186125/128*x^16 + 294775397791875/256*x^14 + 2980506799895625/512*x^12 + 19671344879311125/1024*x^10 + 80473683597181875/2048*x^8 + 187771928393424375/4096*x^6 + 216659917377028125/8192*x^4 + 92854250304440625/16384*x^2 + 6190283353629375/32768

2, 15:

x^30 + 195/2*x^28 + 16695/4*x^26 + 825825/8*x^24 + 26187525/16*x^22 + 559053495/32*x^20 + 8218685475/64*x^18 + 83734375725/128*x^16 + 587583871875/256*x^14 + 2788101941625/512*x^12 + 8662023094725/1024*x^10 + 16709243425875/2048*x^8 + 18535390248375/4096*x^6 + 9587270818125/8192*x^4 + 4108830350625/16384*x^2 - 4108830350625/32768

3, 10:

x^30 + 105/2*x^28 + 4995/4*x^26 + 140175/8*x^24 + 2558925/16*x^22 + 31839885/32*x^20 + 275311575/64*x^18 + 1663875675/128*x^16 + 6981541875/256*x^14 + 19761154875/512*x^12 + 38143776825/1024*x^10 + 43762690125/2048*x^8 - 11345882625/4096*x^6 + 170188239375/8192*x^4 + 510564718125/16384*x^2 + 102112943625/32768

5, 6:

x^30 + 15/2*x^28 + 315/4*x^26 + 2625/8*x^24 + 17325/16*x^22 + 85995/32*x^20 + 694575/64*x^18 - 297675/128*x^16 - 40186125/256*x^14 + 246574125/512*x^12 + 1010606625/1024*x^10 + 7032571875/2048*x^8 + 7657689375/4096*x^6 - 16409334375/8192*x^4 + 49228003125/16384*x^2 + 16409334375/32768

6, 5:

x^30 - 15/2*x^28 + 315/4*x^26 - 2625/8*x^24 + 17325/16*x^22 - 85995/32*x^20 + 694575/64*x^18 + 297675/128*x^16 - 40186125/256*x^14 - 246574125/512*x^12 + 1010606625/1024*x^10 - 7032571875/2048*x^8 + 7657689375/4096*x^6 + 16409334375/8192*x^4 + 49228003125/16384*x^2 - 16409334375/32768

10, 3:

x^30 - 105/2*x^28 + 4995/4*x^26 - 140175/8*x^24 + 2558925/16*x^22 - 31839885/32*x^20 + 275311575/64*x^18 - 1663875675/128*x^16 + 6981541875/256*x^14 - 19761154875/512*x^12 + 38143776825/1024*x^10 - 43762690125/2048*x^8 - 11345882625/4096*x^6 - 170188239375/8192*x^4 + 510564718125/16384*x^2 - 102112943625/32768

15, 2:

x^30 - 195/2*x^28 + 16695/4*x^26 - 825825/8*x^24 + 26187525/16*x^22 - 559053495/32*x^20 + 8218685475/64*x^18 - 83734375725/128*x^16 + 587583871875/256*x^14 - 2788101941625/512*x^12 + 8662023094725/1024*x^10 - 16709243425875/2048*x^8 + 18535390248375/4096*x^6 - 9587270818125/8192*x^4 + 4108830350625/16384*x^2 + 4108830350625/32768

30, 1:

x^30 - 435/2*x^28 + 82215/4*x^26 - 8906625/8*x^24 + 614557125/16*x^22 - 28392539175/32*x^20 + 899097073875/64*x^18 - 19651693186125/128*x^16 + 294775397791875/256*x^14 - 2980506799895625/512*x^12 + 19671344879311125/1024*x^10 - 80473683597181875/2048*x^8 + 187771928393424375/4096*x^6 - 216659917377028125/8192*x^4 + 92854250304440625/16384*x^2 - 6190283353629375/32768

1, 31:

x^31 + 465/2*x^29 + 94395/4*x^27 + 11044215/8*x^25 + 828316125/16*x^23 + 41912795925/32*x^21 + 1466947857375/64*x^19 + 35835440515875/128*x^17 + 609202488769875/256*x^15 + 7107362368981875/512*x^13 + 55437426478058625/1024*x^11 + 277187132390293125/2048*x^9 + 831561397170879375/4096*x^7 + 1343291487737574375/8192*x^5 + 959493919812553125/16384*x^3 + 191898783962510625/32768*x

31, 1:

x^31 - 465/2*x^29 + 94395/4*x^27 - 11044215/8*x^25 + 828316125/16*x^23 - 41912795925/32*x^21 + 1466947857375/64*x^19 - 35835440515875/128*x^17 + 609202488769875/256*x^15 - 7107362368981875/512*x^13 + 55437426478058625/1024*x^11 - 277187132390293125/2048*x^9 + 831561397170879375/4096*x^7 - 1343291487737574375/8192*x^5 + 959493919812553125/16384*x^3 - 191898783962510625/32768*x

1, 32:

x^32 + 248*x^30 + 26970*x^28 + 1699110*x^26 + 276105375/4*x^24 + 3810254175/2*x^22 + 293389571475/8*x^20 + 3981715612875/8*x^18 + 609202488769875/128*x^16 + 1015337481283125/32*x^14 + 18479142159352875/128*x^12 + 55437426478058625/128*x^10 + 831561397170879375/1024*x^8 + 447763829245858125/512*x^6 + 959493919812553125/2048*x^4 + 191898783962510625/2048*x^2 + 191898783962510625/65536

2, 16:

x^32 + 112*x^30 + 5550*x^28 + 160440*x^26 + 12032475/4*x^24 + 38505285*x^22 + 2761844085/8*x^20 + 2190313125*x^18 + 1257458877675/128*x^16 + 492942074625/16*x^14 + 8463544914825/128*x^12 + 3000967944975/32*x^10 + 85372363951875/1024*x^8 + 10865573593875/256*x^6 + 23283371986875/2048*x^4 + 69850115960625/65536

4, 8:

x^32 + 32*x^30 + 510*x^28 + 5040*x^26 + 133875/4*x^24 + 156870*x^22 + 4253445/8*x^20 + 1256850*x^18 + 290828475/128*x^16 + 25302375/8*x^14 + 110437425/128*x^12 - 68762925/16*x^10 + 25786096875/1024*x^8 + 7220107125/128*x^6 + 108301606875/2048*x^4 + 324904820625/65536

8, 4:

x^32 - 32*x^30 + 510*x^28 - 5040*x^26 + 133875/4*x^24 - 156870*x^22 + 4253445/8*x^20 - 1256850*x^18 + 290828475/128*x^16 - 25302375/8*x^14 + 110437425/128*x^12 + 68762925/16*x^10 + 25786096875/1024*x^8 - 7220107125/128*x^6 + 108301606875/2048*x^4 + 324904820625/65536

16, 2:

x^32 - 112*x^30 + 5550*x^28 - 160440*x^26 + 12032475/4*x^24 - 38505285*x^22 + 2761844085/8*x^20 - 2190313125*x^18 + 1257458877675/128*x^16 - 492942074625/16*x^14 + 8463544914825/128*x^12 - 3000967944975/32*x^10 + 85372363951875/1024*x^8 - 10865573593875/256*x^6 + 23283371986875/2048*x^4 + 69850115960625/65536

32, 1:

x^32 - 248*x^30 + 26970*x^28 - 1699110*x^26 + 276105375/4*x^24 - 3810254175/2*x^22 + 293389571475/8*x^20 - 3981715612875/8*x^18 + 609202488769875/128*x^16 - 1015337481283125/32*x^14 + 18479142159352875/128*x^12 - 55437426478058625/128*x^10 + 831561397170879375/1024*x^8 - 447763829245858125/512*x^6 + 959493919812553125/2048*x^4 - 191898783962510625/2048*x^2 + 191898783962510625/65536

1, 33:

x^33 + 264*x^31 + 30690*x^29 + 2076690*x^27 + 364459095/4*x^25 + 5466886425/2*x^23 + 461040755175/8*x^21 + 6915611327625/8*x^19 + 1182569537023875/128*x^17 + 2233742458822875/32*x^15 + 46908591635280375/128*x^13 + 166312279434175875/128*x^11 + 3049058456293224375/1024*x^9 + 2110886623587616875/512*x^7 + 6332659870762850625/2048*x^5 + 2110886623587616875/2048*x^3 + 6332659870762850625/65536*x

3, 11:

x^33 + 66*x^31 + 1980*x^29 + 70785/2*x^27 + 1670625/4*x^25 + 27367065/8*x^23 + 39873735/2*x^21 + 2673957825/32*x^19 + 32247005175/128*x^17 + 69233038875/128*x^15 + 52191060075/64*x^13 + 425632682475/512*x^11 + 540820405125/1024*x^9 + 695340520875/2048*x^7 - 4867383646125/8192*x^3 - 14602150938375/65536*x

11, 3:

x^33 - 66*x^31 + 1980*x^29 - 70785/2*x^27 + 1670625/4*x^25 - 27367065/8*x^23 + 39873735/2*x^21 - 2673957825/32*x^19 + 32247005175/128*x^17 - 69233038875/128*x^15 + 52191060075/64*x^13 - 425632682475/512*x^11 + 540820405125/1024*x^9 - 695340520875/2048*x^7 + 4867383646125/8192*x^3 - 14602150938375/65536*x

33, 1:

x^33 - 264*x^31 + 30690*x^29 - 2076690*x^27 + 364459095/4*x^25 - 5466886425/2*x^23 + 461040755175/8*x^21 - 6915611327625/8*x^19 + 1182569537023875/128*x^17 - 2233742458822875/32*x^15 + 46908591635280375/128*x^13 - 166312279434175875/128*x^11 + 3049058456293224375/1024*x^9 - 2110886623587616875/512*x^7 + 6332659870762850625/2048*x^5 - 2110886623587616875/2048*x^3 + 6332659870762850625/65536*x

1, 34:

x^34 + 561/2*x^32 + 34782*x^30 + 2521695*x^28 + 476600355/4*x^26 + 30979023075/8*x^24 + 712517530725/8*x^22 + 23513078513925/16*x^20 + 2233742458822875/128*x^18 + 37973621799988875/256*x^16 + 113920865399966625/128*x^14 + 942436250126996625/256*x^12 + 10366798751396962875/1024*x^10 + 35885072600989486875/2048*x^8 + 35885072600989486875/2048*x^6 + 35885072600989486875/4096*x^4 + 107655217802968460625/65536*x^2 + 6332659870762850625/131072

2, 17:

x^34 + 255/2*x^32 + 7242*x^30 + 241995*x^28 + 21189735/4*x^26 + 640945305/8*x^24 + 6886663875/8*x^22 + 106527866445/16*x^20 + 4758260782275/128*x^18 + 38136831457725/256*x^16 + 54074384439075/128*x^14 + 211902203638125/256*x^12 + 1106529318469575/1024*x^10 + 1820759689373625/2048*x^8 + 870798112309125/2048*x^6 + 395817323776875/4096*x^4 + 1187451971330625/65536*x^2 - 1187451971330625/131072

17, 2:

x^34 - 255/2*x^32 + 7242*x^30 - 241995*x^28 + 21189735/4*x^26 - 640945305/8*x^24 + 6886663875/8*x^22 - 106527866445/16*x^20 + 4758260782275/128*x^18 - 38136831457725/256*x^16 + 54074384439075/128*x^14 - 211902203638125/256*x^12 + 1106529318469575/1024*x^10 - 1820759689373625/2048*x^8 + 870798112309125/2048*x^6 - 395817323776875/4096*x^4 + 1187451971330625/65536*x^2 + 1187451971330625/131072

34, 1:

x^34 - 561/2*x^32 + 34782*x^30 - 2521695*x^28 + 476600355/4*x^26 - 30979023075/8*x^24 + 712517530725/8*x^22 - 23513078513925/16*x^20 + 2233742458822875/128*x^18 - 37973621799988875/256*x^16 + 113920865399966625/128*x^14 - 942436250126996625/256*x^12 + 10366798751396962875/1024*x^10 - 35885072600989486875/2048*x^8 + 35885072600989486875/2048*x^6 - 35885072600989486875/4096*x^4 + 107655217802968460625/65536*x^2 - 6332659870762850625/131072

1, 35:

x^35 + 595/2*x^33 + 39270*x^31 + 3043425*x^29 + 617815275/4*x^27 + 43370632305/8*x^25 + 1084265807625/8*x^23 + 39188464189875/16*x^21 + 4114788739936875/128*x^19 + 78180986058800625/256*x^17 + 265815352599922125/128*x^15 + 2537328365726529375/256*x^13 + 32985268754444881875/1024*x^11 + 139553060114959115625/2048*x^9 + 179425363004947434375/2048*x^7 + 251195508206926408125/4096*x^5 + 1255977541034632040625/65536*x^3 + 221643095476699771875/131072*x

5, 7:

x^35 + 35/2*x^33 + 210*x^31 + 1575*x^29 + 33075/4*x^27 + 257985/8*x^25 + 826875/8*x^23 + 3142125/16*x^21 + 3472875/128*x^19 + 510512625/256*x^17 + 339647175/128*x^15 - 2406702375/256*x^13 - 94736557125/1024*x^11 - 348971844375/2048*x^9 - 108301606875/2048*x^7 - 758111248125/4096*x^5 + 3790556240625/65536*x^3 + 11371668721875/131072*x

7, 5:

x^35 - 35/2*x^33 + 210*x^31 - 1575*x^29 + 33075/4*x^27 - 257985/8*x^25 + 826875/8*x^23 - 3142125/16*x^21 + 3472875/128*x^19 - 510512625/256*x^17 + 339647175/128*x^15 + 2406702375/256*x^13 - 94736557125/1024*x^11 + 348971844375/2048*x^9 - 108301606875/2048*x^7 + 758111248125/4096*x^5 + 3790556240625/65536*x^3 - 11371668721875/131072*x

35, 1:

x^35 - 595/2*x^33 + 39270*x^31 - 3043425*x^29 + 617815275/4*x^27 - 43370632305/8*x^25 + 1084265807625/8*x^23 - 39188464189875/16*x^21 + 4114788739936875/128*x^19 - 78180986058800625/256*x^17 + 265815352599922125/128*x^15 - 2537328365726529375/256*x^13 + 32985268754444881875/1024*x^11 - 139553060114959115625/2048*x^9 + 179425363004947434375/2048*x^7 - 251195508206926408125/4096*x^5 + 1255977541034632040625/65536*x^3 - 221643095476699771875/131072*x

1, 36:

x^36 + 315*x^34 + 176715/4*x^32 + 3652110*x^30 + 794333925/4*x^28 + 30025822365/4*x^26 + 3252797422875/16*x^24 + 32063288882625/8*x^22 + 7406619731886375/128*x^20 + 78180986058800625/128*x^18 + 2392338173399299125/512*x^16 + 3262279327362680625/128*x^14 + 98955806263334645625/1024*x^12 + 251195508206926408125/1024*x^10 + 1614828267044526909375/4096*x^8 + 753586524620779224375/2048*x^6 + 11303797869311688365625/65536*x^4 + 1994787859290297946875/65536*x^2 + 221643095476699771875/262144

2, 18:

x^36 + 144*x^34 + 37179/4*x^32 + 355572*x^30 + 35955765/4*x^28 + 158609745*x^26 + 32197007115/16*x^24 + 74569569165/4*x^22 + 16234875792135/128*x^20 + 10120147312275/16*x^18 + 1173132426190725/512*x^16 + 379557575171775/64*x^14 + 10948067286381225/1024*x^12 + 3309032826774675/256*x^10 + 40610857419507375/4096*x^8 + 4512317491056375/1024*x^6 + 67684762365845625/65536*x^4 + 22561587455281875/262144

3, 12:

x^36 + 81*x^34 + 11979/4*x^32 + 66528*x^30 + 3948615/4*x^28 + 41257755/4*x^26 + 1249052805/16*x^24 + 434188755*x^22 + 228275041995/128*x^20 + 690524309475/128*x^18 + 6097925658825/512*x^16 + 301615238925/16*x^14 + 21311133418575/1024*x^12 + 16270968188475/1024*x^10 + 27118280314125/4096*x^8 + 569483886596625/65536*x^4 + 569483886596625/65536*x^2 + 189827962198875/262144

4, 9:

x^36 + 45*x^34 + 3915/4*x^32 + 13230*x^30 + 488025/4*x^28 + 3223395/4*x^26 + 62610975/16*x^24 + 112478625/8*x^22 + 4842874575/128*x^20 + 9741414375/128*x^18 + 54607585725/512*x^16 + 13703468625/128*x^14 + 205257331125/1024*x^12 + 34037647875/1024*x^10 - 4595082463125/4096*x^8 - 5003534237625/2048*x^6 - 75053013564375/65536*x^4 - 32165577241875/65536*x^2 + 32165577241875/262144

6, 6:

x^36 + 315/4*x^32 + 4725/4*x^28 + 297675/16*x^24 - 60428025/128*x^20 + 6319937925/512*x^16 + 99112379625/1024*x^12 - 324904820625/4096*x^8 + 34115006165625/65536*x^4 + 11371668721875/262144

9, 4:

x^36 - 45*x^34 + 3915/4*x^32 - 13230*x^30 + 488025/4*x^28 - 3223395/4*x^26 + 62610975/16*x^24 - 112478625/8*x^22 + 4842874575/128*x^20 - 9741414375/128*x^18 + 54607585725/512*x^16 - 13703468625/128*x^14 + 205257331125/1024*x^12 - 34037647875/1024*x^10 - 4595082463125/4096*x^8 + 5003534237625/2048*x^6 - 75053013564375/65536*x^4 + 32165577241875/65536*x^2 + 32165577241875/262144

12, 3:

x^36 - 81*x^34 + 11979/4*x^32 - 66528*x^30 + 3948615/4*x^28 - 41257755/4*x^26 + 1249052805/16*x^24 - 434188755*x^22 + 228275041995/128*x^20 - 690524309475/128*x^18 + 6097925658825/512*x^16 - 301615238925/16*x^14 + 21311133418575/1024*x^12 - 16270968188475/1024*x^10 + 27118280314125/4096*x^8 + 569483886596625/65536*x^4 - 569483886596625/65536*x^2 + 189827962198875/262144

18, 2:

x^36 - 144*x^34 + 37179/4*x^32 - 355572*x^30 + 35955765/4*x^28 - 158609745*x^26 + 32197007115/16*x^24 - 74569569165/4*x^22 + 16234875792135/128*x^20 - 10120147312275/16*x^18 + 1173132426190725/512*x^16 - 379557575171775/64*x^14 + 10948067286381225/1024*x^12 - 3309032826774675/256*x^10 + 40610857419507375/4096*x^8 - 4512317491056375/1024*x^6 + 67684762365845625/65536*x^4 + 22561587455281875/262144

36, 1:

x^36 - 315*x^34 + 176715/4*x^32 - 3652110*x^30 + 794333925/4*x^28 - 30025822365/4*x^26 + 3252797422875/16*x^24 - 32063288882625/8*x^22 + 7406619731886375/128*x^20 - 78180986058800625/128*x^18 + 2392338173399299125/512*x^16 - 3262279327362680625/128*x^14 + 98955806263334645625/1024*x^12 - 251195508206926408125/1024*x^10 + 1614828267044526909375/4096*x^8 - 753586524620779224375/2048*x^6 + 11303797869311688365625/65536*x^4 - 1994787859290297946875/65536*x^2 + 221643095476699771875/262144

1, 37:

x^37 + 333*x^35 + 198135/4*x^33 + 4358970*x^31 + 1013460525/4*x^29 + 41146497315/4*x^27 + 4814140185855/16*x^25 + 51580073419875/8*x^23 + 13049758575228375/128*x^21 + 152247183377664375/128*x^19 + 5206853671516121625/512*x^17 + 8046955674161278875/128*x^15 + 281643448595644760625/1024*x^13 + 844930345786934281875/1024*x^11 + 6638738431183055071875/4096*x^9 + 3983243058709833043125/2048*x^7 + 83648104232906493905625/65536*x^5 + 24602383597913674678125/65536*x^3 + 8200794532637891559375/262144*x

37, 1:

x^37 - 333*x^35 + 198135/4*x^33 - 4358970*x^31 + 1013460525/4*x^29 - 41146497315/4*x^27 + 4814140185855/16*x^25 - 51580073419875/8*x^23 + 13049758575228375/128*x^21 - 152247183377664375/128*x^19 + 5206853671516121625/512*x^17 - 8046955674161278875/128*x^15 + 281643448595644760625/1024*x^13 - 844930345786934281875/1024*x^11 + 6638738431183055071875/4096*x^9 - 3983243058709833043125/2048*x^7 + 83648104232906493905625/65536*x^5 - 24602383597913674678125/65536*x^3 + 8200794532637891559375/262144*x

1, 38:

x^38 + 703/2*x^36 + 221445/4*x^34 + 41410215/8*x^32 + 1283716665/4*x^30 + 111683349855/8*x^28 + 7036051040865/16*x^26 + 326673798325875/32*x^24 + 22540492084485375/128*x^22 + 578539296835124625/256*x^20 + 10992246639867367875/512*x^18 + 152892157809064298625/1024*x^16 + 764460789045321493125/1024*x^14 + 5351225523317250451875/2048*x^12 + 25227206038495609273125/4096*x^10 + 75681618115486827819375/8192*x^8 + 529771326808407794735625/65536*x^6 + 467445288360359818884375/131072*x^4 + 155815096120119939628125/262144*x^2 + 8200794532637891559375/524288

2, 19:

x^38 + 323/2*x^36 + 47025/4*x^34 + 4084335/8*x^32 + 59055705/4*x^30 + 2405208195/8*x^28 + 71165845485/16*x^26 + 1556291011275/32*x^24 + 50711207212575/128*x^22 + 616115118343725/256*x^20 + 5553137682667575/512*x^18 + 36712703954151225/1024*x^16 + 87408934060573125/1024*x^14 + 291842110052868375/2048*x^12 + 657294247863878625/4096*x^10 + 943074355630782375/8192*x^8 + 3172159196212631625/65536*x^6 + 1286010484951066875/131072*x^4 + 428670161650355625/262144*x^2 - 428670161650355625/524288

19, 2:

x^38 - 323/2*x^36 + 47025/4*x^34 - 4084335/8*x^32 + 59055705/4*x^30 - 2405208195/8*x^28 + 71165845485/16*x^26 - 1556291011275/32*x^24 + 50711207212575/128*x^22 - 616115118343725/256*x^20 + 5553137682667575/512*x^18 - 36712703954151225/1024*x^16 + 87408934060573125/1024*x^14 - 291842110052868375/2048*x^12 + 657294247863878625/4096*x^10 - 943074355630782375/8192*x^8 + 3172159196212631625/65536*x^6 - 1286010484951066875/131072*x^4 + 428670161650355625/262144*x^2 + 428670161650355625/524288

38, 1:

x^38 - 703/2*x^36 + 221445/4*x^34 - 41410215/8*x^32 + 1283716665/4*x^30 - 111683349855/8*x^28 + 7036051040865/16*x^26 - 326673798325875/32*x^24 + 22540492084485375/128*x^22 - 578539296835124625/256*x^20 + 10992246639867367875/512*x^18 - 152892157809064298625/1024*x^16 + 764460789045321493125/1024*x^14 - 5351225523317250451875/2048*x^12 + 25227206038495609273125/4096*x^10 - 75681618115486827819375/8192*x^8 + 529771326808407794735625/65536*x^6 - 467445288360359818884375/131072*x^4 + 155815096120119939628125/262144*x^2 - 8200794532637891559375/524288

1, 39:

x^39 + 741/2*x^37 + 246753/4*x^35 + 48939345/8*x^33 + 1614998385/4*x^31 + 150194849805/8*x^29 + 10163184836805/16*x^27 + 509611125388365/32*x^25 + 38220834404127375/128*x^23 + 1074430122693802875/256*x^21 + 22563032576569860375/512*x^19 + 350752597326676920375/1024*x^17 + 1987598051517835882125/1024*x^15 + 16053676569951751355625/2048*x^13 + 89441912318302614695625/4096*x^11 + 327953678500442920550625/8192*x^9 + 2951583106503986284955625/65536*x^7 + 3646073249210806587298125/131072*x^5 + 2025596249561559215165625/262144*x^3 + 319830986772877770815625/524288*x

3, 13:

x^39 + 195/2*x^37 + 17433/4*x^35 + 943215/8*x^33 + 8603595/4*x^31 + 223648425/8*x^29 + 4270941675/16*x^27 + 60950614725/32*x^25 + 1310793959475/128*x^23 + 10636580579625/256*x^21 + 64870822291275/512*x^19 + 294466175128125/1024*x^17 + 489608224580475/1024*x^15 + 1167729585647625/2048*x^13 + 1912249645180875/4096*x^11 + 2076055015159125/8192*x^9 + 8813441102090625/65536*x^7 - 7403290525756125/131072*x^5 - 61694087714634375/262144*x^3 - 37016452628780625/524288*x

13, 3:

x^39 - 195/2*x^37 + 17433/4*x^35 - 943215/8*x^33 + 8603595/4*x^31 - 223648425/8*x^29 + 4270941675/16*x^27 - 60950614725/32*x^25 + 1310793959475/128*x^23 - 10636580579625/256*x^21 + 64870822291275/512*x^19 - 294466175128125/1024*x^17 + 489608224580475/1024*x^15 - 1167729585647625/2048*x^13 + 1912249645180875/4096*x^11 - 2076055015159125/8192*x^9 + 8813441102090625/65536*x^7 + 7403290525756125/131072*x^5 - 61694087714634375/262144*x^3 + 37016452628780625/524288*x

39, 1:

x^39 - 741/2*x^37 + 246753/4*x^35 - 48939345/8*x^33 + 1614998385/4*x^31 - 150194849805/8*x^29 + 10163184836805/16*x^27 - 509611125388365/32*x^25 + 38220834404127375/128*x^23 - 1074430122693802875/256*x^21 + 22563032576569860375/512*x^19 - 350752597326676920375/1024*x^17 + 1987598051517835882125/1024*x^15 - 16053676569951751355625/2048*x^13 + 89441912318302614695625/4096*x^11 - 327953678500442920550625/8192*x^9 + 2951583106503986284955625/65536*x^7 - 3646073249210806587298125/131072*x^5 + 2025596249561559215165625/262144*x^3 - 319830986772877770815625/524288*x

1, 40:

x^40 + 390*x^38 + 137085/2*x^36 + 14393925/2*x^34 + 8074991925/16*x^32 + 50064949935/2*x^30 + 7259417740575/8*x^28 + 196004278995525/8*x^26 + 63701390673545625/128*x^24 + 488377328497183125/64*x^22 + 22563032576569860375/256*x^20 + 194862554070376066875/256*x^18 + 9937990257589179410625/2048*x^16 + 11466911835679822396875/512*x^14 + 149069853863837691159375/2048*x^12 + 327953678500442920550625/2048*x^10 + 14757915532519931424778125/65536*x^8 + 6076788748684677645496875/32768*x^6 + 10127981247807796075828125/131072*x^4 + 1599154933864388854078125/131072*x^2 + 319830986772877770815625/1048576

2, 20:

x^40 + 180*x^38 + 29355/2*x^36 + 718200*x^34 + 376965225/16*x^32 + 548507295*x^30 + 74884828725/8*x^28 + 238600673325/2*x^26 + 146770319138625/128*x^24 + 266804181291375/32*x^22 + 11714288866104825/256*x^20 + 6020204381816625/32*x^18 + 1171298749385890125/2048*x^16 + 322751660170314375/256*x^14 + 4013911513635148125/2048*x^12 + 1057386398737543875/512*x^10 + 91306744431525748125/65536*x^8 + 9002073394657468125/16384*x^6 + 15003455657762446875/131072*x^4 + 9002073394657468125/1048576

4, 10:

x^40 + 60*x^38 + 3435/2*x^36 + 30600*x^34 + 6006825/16*x^32 + 3339765*x^30 + 177405525/8*x^28 + 223128675/2*x^26 + 54905090625/128*x^24 + 40413421125/32*x^22 + 722786751225/256*x^20 + 152899275375/32*x^18 + 13010485690125/2048*x^16 + 1288568098125/256*x^14 - 2212447111875/2048*x^12 + 4867383646125/512*x^10 + 4599677545588125/65536*x^8 + 1533225848529375/16384*x^6 + 7666129242646875/131072*x^4 + 4599677545588125/1048576

5, 8:

x^40 + 30*x^38 + 1005/2*x^36 + 11025/2*x^34 + 685125/16*x^32 + 494235/2*x^30 + 8783775/8*x^28 + 29469825/8*x^26 + 1165397625/128*x^24 + 1427351625/64*x^22 + 9007943175/256*x^20 - 5844848625/256*x^18 - 434237871375/2048*x^16 + 541508034375/512*x^14 + 13537700859375/2048*x^12 + 21205454626125/2048*x^10 + 997132894498125/65536*x^8 + 75053013564375/32768*x^6 - 375265067821875/131072*x^4 + 1125795203465625/131072*x^2 + 1125795203465625/1048576

8, 5:

x^40 - 30*x^38 + 1005/2*x^36 - 11025/2*x^34 + 685125/16*x^32 - 494235/2*x^30 + 8783775/8*x^28 - 29469825/8*x^26 + 1165397625/128*x^24 - 1427351625/64*x^22 + 9007943175/256*x^20 + 5844848625/256*x^18 - 434237871375/2048*x^16 - 541508034375/512*x^14 + 13537700859375/2048*x^12 - 21205454626125/2048*x^10 + 997132894498125/65536*x^8 - 75053013564375/32768*x^6 - 375265067821875/131072*x^4 - 1125795203465625/131072*x^2 + 1125795203465625/1048576

10, 4:

x^40 - 60*x^38 + 3435/2*x^36 - 30600*x^34 + 6006825/16*x^32 - 3339765*x^30 + 177405525/8*x^28 - 223128675/2*x^26 + 54905090625/128*x^24 - 40413421125/32*x^22 + 722786751225/256*x^20 - 152899275375/32*x^18 + 13010485690125/2048*x^16 - 1288568098125/256*x^14 - 2212447111875/2048*x^12 - 4867383646125/512*x^10 + 4599677545588125/65536*x^8 - 1533225848529375/16384*x^6 + 7666129242646875/131072*x^4 + 4599677545588125/1048576

20, 2:

x^40 - 180*x^38 + 29355/2*x^36 - 718200*x^34 + 376965225/16*x^32 - 548507295*x^30 + 74884828725/8*x^28 - 238600673325/2*x^26 + 146770319138625/128*x^24 - 266804181291375/32*x^22 + 11714288866104825/256*x^20 - 6020204381816625/32*x^18 + 1171298749385890125/2048*x^16 - 322751660170314375/256*x^14 + 4013911513635148125/2048*x^12 - 1057386398737543875/512*x^10 + 91306744431525748125/65536*x^8 - 9002073394657468125/16384*x^6 + 15003455657762446875/131072*x^4 + 9002073394657468125/1048576

40, 1:

x^40 - 390*x^38 + 137085/2*x^36 - 14393925/2*x^34 + 8074991925/16*x^32 - 50064949935/2*x^30 + 7259417740575/8*x^28 - 196004278995525/8*x^26 + 63701390673545625/128*x^24 - 488377328497183125/64*x^22 + 22563032576569860375/256*x^20 - 194862554070376066875/256*x^18 + 9937990257589179410625/2048*x^16 - 11466911835679822396875/512*x^14 + 149069853863837691159375/2048*x^12 - 327953678500442920550625/2048*x^10 + 14757915532519931424778125/65536*x^8 - 6076788748684677645496875/32768*x^6 + 10127981247807796075828125/131072*x^4 - 1599154933864388854078125/131072*x^2 + 319830986772877770815625/1048576

1, 41:

x^41 + 410*x^39 + 151905/2*x^37 + 16861455/2*x^35 + 10032565725/16*x^33 + 66214933785/2*x^31 + 10263314736675/8*x^29 + 297636127363575/8*x^27 + 104470280704614825/128*x^25 + 870585672538456875/64*x^23 + 44051635030445917875/256*x^21 + 420492879836074670625/256*x^19 + 23968094150656256225625/2048*x^17 + 31342892350858181218125/512*x^15 + 470143385262872718271875/2048*x^13 + 1222372801683469067506875/2048*x^11 + 67230504092590798712878125/65536*x^9 + 35592619813724540495053125/32768*x^7 + 83049446232023927821790625/131072*x^5 + 21855117429479981005734375/131072*x^3 + 13113070457687988603440625/1048576*x

41, 1:

x^41 - 410*x^39 + 151905/2*x^37 - 16861455/2*x^35 + 10032565725/16*x^33 - 66214933785/2*x^31 + 10263314736675/8*x^29 - 297636127363575/8*x^27 + 104470280704614825/128*x^25 - 870585672538456875/64*x^23 + 44051635030445917875/256*x^21 - 420492879836074670625/256*x^19 + 23968094150656256225625/2048*x^17 - 31342892350858181218125/512*x^15 + 470143385262872718271875/2048*x^13 - 1222372801683469067506875/2048*x^11 + 67230504092590798712878125/65536*x^9 - 35592619813724540495053125/32768*x^7 + 83049446232023927821790625/131072*x^5 - 21855117429479981005734375/131072*x^3 + 13113070457687988603440625/1048576*x

1, 42:

x^42 + 861/2*x^40 + 167895/2*x^38 + 39343395/4*x^36 + 12393169425/16*x^34 + 1390513609485/32*x^32 + 14368640631345/8*x^30 + 892908382090725/16*x^28 + 168759684215147025/128*x^26 + 6094099707769198125/256*x^24 + 84098575967214934125/256*x^22 + 1766070095311513616625/512*x^20 + 55925553018197931193125/2048*x^18 + 658200739368021805580625/4096*x^16 + 1410430155788618154815625/2048*x^14 + 8556609611784283472548125/4096*x^12 + 282368117188881354594088125/65536*x^10 + 747445016088215350396115625/131072*x^8 + 581346123624167494752534375/131072*x^6 + 458957466019079601120421875/262144*x^4 + 275374479611447760672253125/1048576*x^2 + 13113070457687988603440625/2097152

2, 21:

x^42 + 399/2*x^40 + 36225/2*x^38 + 3968055/4*x^36 + 586320525/16*x^34 + 30953576115/32*x^32 + 150913372995/8*x^30 + 4435789483125/16*x^28 + 397430168006025/128*x^26 + 6817435239990375/256*x^24 + 44775193609221075/256*x^22 + 448278933417239925/512*x^20 + 6778607759814515625/2048*x^18 + 38152843464256896375/4096*x^16 + 39152873995211851875/2048*x^14 + 113898960950989339125/4096*x^12 + 1804615646515667110125/65536*x^10 + 2295528715637654371875/131072*x^8 + 861198354755564450625/131072*x^6 + 315072568813011384375/262144*x^4 + 189043541287806830625/1048576*x^2 - 189043541287806830625/2097152

3, 14:

x^42 + 231/2*x^40 + 12285/2*x^38 + 795795/4*x^36 + 70085925/16*x^34 + 2220132915/32*x^32 + 6532380855/8*x^30 + 116461370025/16*x^28 + 6357417291225/128*x^26 + 66708774507375/256*x^24 + 268972058730375/256*x^22 + 1658597851535625/512*x^20 + 15493242692552625/2048*x^18 + 54024271060134375/4096*x^16 + 34411454186484375/2048*x^14 + 62342187221939625/4096*x^12 + 625989343344490125/65536*x^10 + 407180978916586875/131072*x^8 + 86371722800488125/131072*x^6 + 1295575842007321875/262144*x^4 + 3886727526021965625/1048576*x^2 + 555246789431709375/2097152

6, 7:

x^42 + 21/2*x^40 + 315/2*x^38 + 4095/4*x^36 + 99225/16*x^34 + 813645/32*x^32 + 866565/8*x^30 + 4068225/16*x^28 - 56260575/128*x^26 + 552187125/256*x^24 + 9895610025/256*x^22 - 38988578475/512*x^20 - 551134843875/2048*x^18 - 12498005433375/4096*x^16 - 11371668721875/2048*x^14 - 11978157720375/4096*x^12 - 295208520019875/65536*x^10 - 4278021773169375/131072*x^8 + 5779082044456875/131072*x^6 + 7880566424259375/262144*x^4 + 23641699272778125/1048576*x^2 - 7880566424259375/2097152

7, 6:

x^42 - 21/2*x^40 + 315/2*x^38 - 4095/4*x^36 + 99225/16*x^34 - 813645/32*x^32 + 866565/8*x^30 - 4068225/16*x^28 - 56260575/128*x^26 - 552187125/256*x^24 + 9895610025/256*x^22 + 38988578475/512*x^20 - 551134843875/2048*x^18 + 12498005433375/4096*x^16 - 11371668721875/2048*x^14 + 11978157720375/4096*x^12 - 295208520019875/65536*x^10 + 4278021773169375/131072*x^8 + 5779082044456875/131072*x^6 - 7880566424259375/262144*x^4 + 23641699272778125/1048576*x^2 + 7880566424259375/2097152

14, 3:

x^42 - 231/2*x^40 + 12285/2*x^38 - 795795/4*x^36 + 70085925/16*x^34 - 2220132915/32*x^32 + 6532380855/8*x^30 - 116461370025/16*x^28 + 6357417291225/128*x^26 - 66708774507375/256*x^24 + 268972058730375/256*x^22 - 1658597851535625/512*x^20 + 15493242692552625/2048*x^18 - 54024271060134375/4096*x^16 + 34411454186484375/2048*x^14 - 62342187221939625/4096*x^12 + 625989343344490125/65536*x^10 - 407180978916586875/131072*x^8 + 86371722800488125/131072*x^6 - 1295575842007321875/262144*x^4 + 3886727526021965625/1048576*x^2 - 555246789431709375/2097152

21, 2:

x^42 - 399/2*x^40 + 36225/2*x^38 - 3968055/4*x^36 + 586320525/16*x^34 - 30953576115/32*x^32 + 150913372995/8*x^30 - 4435789483125/16*x^28 + 397430168006025/128*x^26 - 6817435239990375/256*x^24 + 44775193609221075/256*x^22 - 448278933417239925/512*x^20 + 6778607759814515625/2048*x^18 - 38152843464256896375/4096*x^16 + 39152873995211851875/2048*x^14 - 113898960950989339125/4096*x^12 + 1804615646515667110125/65536*x^10 - 2295528715637654371875/131072*x^8 + 861198354755564450625/131072*x^6 - 315072568813011384375/262144*x^4 + 189043541287806830625/1048576*x^2 + 189043541287806830625/2097152

42, 1:

x^42 - 861/2*x^40 + 167895/2*x^38 - 39343395/4*x^36 + 12393169425/16*x^34 - 1390513609485/32*x^32 + 14368640631345/8*x^30 - 892908382090725/16*x^28 + 168759684215147025/128*x^26 - 6094099707769198125/256*x^24 + 84098575967214934125/256*x^22 - 1766070095311513616625/512*x^20 + 55925553018197931193125/2048*x^18 - 658200739368021805580625/4096*x^16 + 1410430155788618154815625/2048*x^14 - 8556609611784283472548125/4096*x^12 + 282368117188881354594088125/65536*x^10 - 747445016088215350396115625/131072*x^8 + 581346123624167494752534375/131072*x^6 - 458957466019079601120421875/262144*x^4 + 275374479611447760672253125/1048576*x^2 - 13113070457687988603440625/2097152

1, 43:

x^43 + 903/2*x^41 + 185115/2*x^39 + 45723405/4*x^37 + 15225893865/16*x^35 + 1811881369935/32*x^33 + 19930695069285/8*x^31 + 1323967601031075/16*x^29 + 268765423009308225/128*x^27 + 10481851497363020775/256*x^25 + 157227772460445311625/256*x^23 + 3616238766590242167375/512*x^21 + 126568356830658475858125/2048*x^19 + 1664860693695584567056875/4096*x^17 + 4043233113260705377138125/2048*x^15 + 28302631792824937639966875/4096*x^13 + 1103802639920172567958708125/65536*x^11 + 3571126187977028896336996875/131072*x^9 + 3571126187977028896336996875/131072*x^7 + 3947034207764084569635628125/262144*x^5 + 3947034207764084569635628125/1048576*x^3 + 563862029680583509947946875/2097152*x

43, 1:

x^43 - 903/2*x^41 + 185115/2*x^39 - 45723405/4*x^37 + 15225893865/16*x^35 - 1811881369935/32*x^33 + 19930695069285/8*x^31 - 1323967601031075/16*x^29 + 268765423009308225/128*x^27 - 10481851497363020775/256*x^25 + 157227772460445311625/256*x^23 - 3616238766590242167375/512*x^21 + 126568356830658475858125/2048*x^19 - 1664860693695584567056875/4096*x^17 + 4043233113260705377138125/2048*x^15 - 28302631792824937639966875/4096*x^13 + 1103802639920172567958708125/65536*x^11 - 3571126187977028896336996875/131072*x^9 + 3571126187977028896336996875/131072*x^7 - 3947034207764084569635628125/262144*x^5 + 3947034207764084569635628125/1048576*x^3 - 563862029680583509947946875/2097152*x

1, 44:

x^44 + 473*x^42 + 407253/4*x^40 + 26471445/2*x^38 + 18609425835/16*x^36 + 1172393827605/16*x^34 + 219237645762135/64*x^32 + 970909574089455/8*x^30 + 422345664728912925/128*x^28 + 8869258959307171425/128*x^26 + 576501832354966142625/512*x^24 + 3616238766590242167375/256*x^22 + 278450385027448646887875/2048*x^20 + 2034829736739047804180625/2048*x^18 + 44475564245867759148519375/8192*x^16 + 44475564245867759148519375/2048*x^14 + 4047276346373966082515263125/65536*x^12 + 7856477613549463571941393125/65536*x^10 + 39282388067747317859706965625/262144*x^8 + 14472458761801643421997303125/131072*x^6 + 43417376285404930265991909375/1048576*x^4 + 6202482326486418609427415625/1048576*x^2 + 563862029680583509947946875/4194304

2, 22:

x^44 + 220*x^42 + 88473/4*x^40 + 1347885*x^38 + 891208395/16*x^36 + 1654894395*x^34 + 2341025777475/64*x^32 + 2456479695285/4*x^30 + 1014882595667325/128*x^28 + 2535082724322225/32*x^26 + 314159285180265525/512*x^24 + 470403031559335875/128*x^22 + 34637210141950491075/2048*x^20 + 15149492310117540375/256*x^18 + 1270044170160982090875/8192*x^16 + 304832847720871094625/1024*x^14 + 26663141164598236423125/65536*x^12 + 6225233821518794457375/16384*x^10 + 59974813646339605138125/262144*x^8 + 5314223993979458683125/65536*x^6 + 15942671981938376049375/1048576*x^4 + 4348001449619557104375/4194304

4, 11:

x^44 + 77*x^42 + 11253/4*x^40 + 128205/2*x^38 + 16202835/16*x^36 + 187390665/16*x^34 + 6548652495/64*x^32 + 5503081815/8*x^30 + 460399279725/128*x^28 + 1882032369525/128*x^26 + 24059718507825/512*x^24 + 29972286919575/256*x^22 + 463761223400475/2048*x^20 + 679193168779125/2048*x^18 + 2940595062780375/8192*x^16 + 760007189316375/2048*x^14 + 25120566997651125/65536*x^12 - 64731335109816375/65536*x^10 - 1033613254172874375/262144*x^8 - 657753889019101875/131072*x^6 - 1973261667057305625/1048576*x^4 - 657753889019101875/1048576*x^2 + 657753889019101875/4194304

11, 4:

x^44 - 77*x^42 + 11253/4*x^40 - 128205/2*x^38 + 16202835/16*x^36 - 187390665/16*x^34 + 6548652495/64*x^32 - 5503081815/8*x^30 + 460399279725/128*x^28 - 1882032369525/128*x^26 + 24059718507825/512*x^24 - 29972286919575/256*x^22 + 463761223400475/2048*x^20 - 679193168779125/2048*x^18 + 2940595062780375/8192*x^16 - 760007189316375/2048*x^14 + 25120566997651125/65536*x^12 + 64731335109816375/65536*x^10 - 1033613254172874375/262144*x^8 + 657753889019101875/131072*x^6 - 1973261667057305625/1048576*x^4 + 657753889019101875/1048576*x^2 + 657753889019101875/4194304

22, 2:

x^44 - 220*x^42 + 88473/4*x^40 - 1347885*x^38 + 891208395/16*x^36 - 1654894395*x^34 + 2341025777475/64*x^32 - 2456479695285/4*x^30 + 1014882595667325/128*x^28 - 2535082724322225/32*x^26 + 314159285180265525/512*x^24 - 470403031559335875/128*x^22 + 34637210141950491075/2048*x^20 - 15149492310117540375/256*x^18 + 1270044170160982090875/8192*x^16 - 304832847720871094625/1024*x^14 + 26663141164598236423125/65536*x^12 - 6225233821518794457375/16384*x^10 + 59974813646339605138125/262144*x^8 - 5314223993979458683125/65536*x^6 + 15942671981938376049375/1048576*x^4 + 4348001449619557104375/4194304

44, 1:

x^44 - 473*x^42 + 407253/4*x^40 - 26471445/2*x^38 + 18609425835/16*x^36 - 1172393827605/16*x^34 + 219237645762135/64*x^32 - 970909574089455/8*x^30 + 422345664728912925/128*x^28 - 8869258959307171425/128*x^26 + 576501832354966142625/512*x^24 - 3616238766590242167375/256*x^22 + 278450385027448646887875/2048*x^20 - 2034829736739047804180625/2048*x^18 + 44475564245867759148519375/8192*x^16 - 44475564245867759148519375/2048*x^14 + 4047276346373966082515263125/65536*x^12 - 7856477613549463571941393125/65536*x^10 + 39282388067747317859706965625/262144*x^8 - 14472458761801643421997303125/131072*x^6 + 43417376285404930265991909375/1048576*x^4 - 6202482326486418609427415625/1048576*x^2 + 563862029680583509947946875/4194304

1, 45:

x^45 + 495*x^43 + 446985/4*x^41 + 30543975/2*x^39 + 22633085475/16*x^37 + 1507363492635/16*x^35 + 298960426039275/64*x^33 + 1409384865613725/8*x^31 + 655363962510382125/128*x^29 + 14782098265511952375/128*x^27 + 1037703298238939056725/512*x^25 + 7075249760720039023125/256*x^23 + 596679396487389957616875/2048*x^21 + 4819333587013534273059375/2048*x^19 + 117729434768473480099021875/8192*x^17 + 133426692737603277445558125/2048*x^15 + 14009802737448344131783603125/65536*x^13 + 32140135691793260067032971875/65536*x^11 + 196411940338736589298534828125/262144*x^9 + 93037234897296279141411234375/131072*x^7 + 390756386568644372393927184375/1048576*x^5 + 93037234897296279141411234375/1048576*x^3 + 25373791335626257947657609375/4194304*x

3, 15:

x^45 + 135*x^43 + 33705/4*x^41 + 644175/2*x^39 + 134827875/16*x^37 + 2558216115/16*x^35 + 145646721675/64*x^33 + 198431558325/8*x^31 + 26828010334125/128*x^29 + 176908339983375/128*x^27 + 3648905312577525/512*x^25 + 7343392909978125/256*x^23 + 183565648207816875/2048*x^21 + 441196615779294375/2048*x^19 + 3218639928671041875/8192*x^17 + 1093140344760589125/2048*x^15 + 34525305766669843125/65536*x^13 + 23702868499962526875/65536*x^11 + 45098378119397728125/262144*x^9 + 9439195420339059375/131072*x^7 - 66074367942373415625/1048576*x^5 - 141587931305085890625/1048576*x^3 - 141587931305085890625/4194304*x

5, 9:

x^45 + 45*x^43 + 4185/4*x^41 + 31725/2*x^39 + 2743875/16*x^37 + 22199265/16*x^35 + 553916475/64*x^33 + 337265775/8*x^31 + 20635894125/128*x^29 + 63540217125/128*x^27 + 615948812325/512*x^25 + 541508034375/256*x^23 + 6037630396875/2048*x^21 + 19571647528125/2048*x^19 + 154701109591875/8192*x^17 - 208412517938625/2048*x^15 - 33570140781436875/65536*x^13 - 80703433299864375/65536*x^11 - 314311298948521875/262144*x^9 - 68995163183821875/131072*x^7 - 869339056116155625/1048576*x^5 + 482966142286753125/1048576*x^3 + 1448898426860259375/4194304*x

9, 5:

x^45 - 45*x^43 + 4185/4*x^41 - 31725/2*x^39 + 2743875/16*x^37 - 22199265/16*x^35 + 553916475/64*x^33 - 337265775/8*x^31 + 20635894125/128*x^29 - 63540217125/128*x^27 + 615948812325/512*x^25 - 541508034375/256*x^23 + 6037630396875/2048*x^21 - 19571647528125/2048*x^19 + 154701109591875/8192*x^17 + 208412517938625/2048*x^15 - 33570140781436875/65536*x^13 + 80703433299864375/65536*x^11 - 314311298948521875/262144*x^9 + 68995163183821875/131072*x^7 - 869339056116155625/1048576*x^5 - 482966142286753125/1048576*x^3 + 1448898426860259375/4194304*x

15, 3:

x^45 - 135*x^43 + 33705/4*x^41 - 644175/2*x^39 + 134827875/16*x^37 - 2558216115/16*x^35 + 145646721675/64*x^33 - 198431558325/8*x^31 + 26828010334125/128*x^29 - 176908339983375/128*x^27 + 3648905312577525/512*x^25 - 7343392909978125/256*x^23 + 183565648207816875/2048*x^21 - 441196615779294375/2048*x^19 + 3218639928671041875/8192*x^17 - 1093140344760589125/2048*x^15 + 34525305766669843125/65536*x^13 - 23702868499962526875/65536*x^11 + 45098378119397728125/262144*x^9 - 9439195420339059375/131072*x^7 - 66074367942373415625/1048576*x^5 + 141587931305085890625/1048576*x^3 - 141587931305085890625/4194304*x

45, 1:

x^45 - 495*x^43 + 446985/4*x^41 - 30543975/2*x^39 + 22633085475/16*x^37 - 1507363492635/16*x^35 + 298960426039275/64*x^33 - 1409384865613725/8*x^31 + 655363962510382125/128*x^29 - 14782098265511952375/128*x^27 + 1037703298238939056725/512*x^25 - 7075249760720039023125/256*x^23 + 596679396487389957616875/2048*x^21 - 4819333587013534273059375/2048*x^19 + 117729434768473480099021875/8192*x^17 - 133426692737603277445558125/2048*x^15 + 14009802737448344131783603125/65536*x^13 - 32140135691793260067032971875/65536*x^11 + 196411940338736589298534828125/262144*x^9 - 93037234897296279141411234375/131072*x^7 + 390756386568644372393927184375/1048576*x^5 - 93037234897296279141411234375/1048576*x^3 + 25373791335626257947657609375/4194304*x

1, 46:

x^46 + 1035/2*x^44 + 489555/4*x^42 + 140502285/8*x^40 + 27397945575/16*x^38 + 3852151147845/32*x^36 + 404475870523725/64*x^34 + 32415851909115675/128*x^32 + 1004891409182585925/128*x^30 + 48569751443824986375/256*x^28 + 1835936604576584484975/512*x^26 + 54243581498853632510625/1024*x^24 + 1247602374473633547744375/2048*x^22 + 22168934500262257656073125/4096*x^20 + 300864111074987782475278125/8192*x^18 + 3068813932964875381247836875/16384*x^16 + 46032208994473130718717553125/65536*x^14 + 246407706970414993847252784375/131072*x^12 + 903494925558188310773260209375/262144*x^10 + 2139856402637814420252458390625/524288*x^8 + 2995798963692940188353441746875/1048576*x^6 + 2139856402637814420252458390625/2097152*x^4 + 583597200719403932796125015625/4194304*x^2 + 25373791335626257947657609375/8388608

2, 23:

x^46 + 483/2*x^44 + 107019/4*x^42 + 14435421/8*x^40 + 1326842055/16*x^38 + 88164586125/32*x^36 + 4385272444245/64*x^34 + 166841658865035/128*x^32 + 2460137841221445/128*x^30 + 56660529791440575/256*x^28 + 1022275495055759175/512*x^26 + 14438510043055923825/1024*x^24 + 158977415376450645975/2048*x^22 + 1354157150277186780525/4096*x^20 + 8820681604979097979125/8192*x^18 + 43233326908862658442875/16384*x^16 + 311964439749098188177125/65536*x^14 + 804159417312335801091375/131072*x^12 + 1421327165932620363792375/262144*x^10 + 1623875017588866017600625/524288*x^8 + 1100044366753747947406875/1048576*x^6 + 366681455584582649135625/2097152*x^4 + 100004033341249813400625/4194304*x^2 - 100004033341249813400625/8388608

23, 2:

x^46 - 483/2*x^44 + 107019/4*x^42 - 14435421/8*x^40 + 1326842055/16*x^38 - 88164586125/32*x^36 + 4385272444245/64*x^34 - 166841658865035/128*x^32 + 2460137841221445/128*x^30 - 56660529791440575/256*x^28 + 1022275495055759175/512*x^26 - 14438510043055923825/1024*x^24 + 158977415376450645975/2048*x^22 - 1354157150277186780525/4096*x^20 + 8820681604979097979125/8192*x^18 - 43233326908862658442875/16384*x^16 + 311964439749098188177125/65536*x^14 - 804159417312335801091375/131072*x^12 + 1421327165932620363792375/262144*x^10 - 1623875017588866017600625/524288*x^8 + 1100044366753747947406875/1048576*x^6 - 366681455584582649135625/2097152*x^4 + 100004033341249813400625/4194304*x^2 + 100004033341249813400625/8388608

46, 1:

x^46 - 1035/2*x^44 + 489555/4*x^42 - 140502285/8*x^40 + 27397945575/16*x^38 - 3852151147845/32*x^36 + 404475870523725/64*x^34 - 32415851909115675/128*x^32 + 1004891409182585925/128*x^30 - 48569751443824986375/256*x^28 + 1835936604576584484975/512*x^26 - 54243581498853632510625/1024*x^24 + 1247602374473633547744375/2048*x^22 - 22168934500262257656073125/4096*x^20 + 300864111074987782475278125/8192*x^18 - 3068813932964875381247836875/16384*x^16 + 46032208994473130718717553125/65536*x^14 - 246407706970414993847252784375/131072*x^12 + 903494925558188310773260209375/262144*x^10 - 2139856402637814420252458390625/524288*x^8 + 2995798963692940188353441746875/1048576*x^6 - 2139856402637814420252458390625/2097152*x^4 + 583597200719403932796125015625/4194304*x^2 - 25373791335626257947657609375/8388608

1, 47:

x^47 + 1081/2*x^45 + 535095/4*x^43 + 161063595/8*x^41 + 33018036975/16*x^39 + 4893273079695/32*x^37 + 543153311846145/64*x^35 + 46168031506922325/128*x^33 + 1523545039728436725/128*x^31 + 78716493719302564125/256*x^29 + 3195889645003684103475/512*x^27 + 101977933217844829119975/1024*x^25 + 2549448330446120727999375/2048*x^23 + 49616186738682195706449375/4096*x^21 + 744242801080232935596740625/8192*x^19 + 8484367932314655465802843125/16384*x^17 + 144234254849349142918648333125/65536*x^15 + 890858632893038823909298528125/131072*x^13 + 3860387409203168236940293621875/262144*x^11 + 11174805658219697527985060484375/524288*x^9 + 20114650184795455550373108871875/1048576*x^7 + 20114650184795455550373108871875/2097152*x^5 + 9143022811270661613805958578125/4194304*x^3 + 1192568192774434123539907640625/8388608*x

47, 1:

x^47 - 1081/2*x^45 + 535095/4*x^43 - 161063595/8*x^41 + 33018036975/16*x^39 - 4893273079695/32*x^37 + 543153311846145/64*x^35 - 46168031506922325/128*x^33 + 1523545039728436725/128*x^31 - 78716493719302564125/256*x^29 + 3195889645003684103475/512*x^27 - 101977933217844829119975/1024*x^25 + 2549448330446120727999375/2048*x^23 - 49616186738682195706449375/4096*x^21 + 744242801080232935596740625/8192*x^19 - 8484367932314655465802843125/16384*x^17 + 144234254849349142918648333125/65536*x^15 - 890858632893038823909298528125/131072*x^13 + 3860387409203168236940293621875/262144*x^11 - 11174805658219697527985060484375/524288*x^9 + 20114650184795455550373108871875/1048576*x^7 - 20114650184795455550373108871875/2097152*x^5 + 9143022811270661613805958578125/4194304*x^3 - 1192568192774434123539907640625/8388608*x

1, 48:

x^48 + 564*x^46 + 145935*x^44 + 23009085*x^42 + 19810822185/8*x^40 + 772622065215/4*x^38 + 181051103948715/16*x^36 + 8147299677692175/16*x^34 + 4570635119185310175/256*x^32 + 15743298743860512825/32*x^30 + 1369666990715864615775/128*x^28 + 23533369204118037489225/128*x^26 + 2549448330446120727999375/1024*x^24 + 13531687292367871556304375/512*x^22 + 446545680648139761358044375/2048*x^20 + 2828122644104885155267614375/2048*x^18 + 432702764548047428755944999375/65536*x^16 + 381796556954159495961127940625/16384*x^14 + 3860387409203168236940293621875/65536*x^12 + 6704883394931818516791036290625/65536*x^10 + 60343950554386366651119326615625/524288*x^8 + 20114650184795455550373108871875/262144*x^6 + 27429068433811984841417875734375/1048576*x^4 + 3577704578323302370619722921875/1048576*x^2 + 1192568192774434123539907640625/16777216

2, 24:

x^48 + 264*x^46 + 32085*x^44 + 2383260*x^42 + 969330285/8*x^40 + 8950625145/2*x^38 + 1990744998165/16*x^36 + 2662403938425*x^34 + 11384613336425175/256*x^32 + 9356259740976225/16*x^30 + 778081248058254525/128*x^28 + 1599697585314276525/32*x^26 + 332239535893121731875/1024*x^24 + 423123916284292348125/256*x^22 + 13417351864650579470625/2048*x^20 + 5109144223069772510625/256*x^18 + 3011093010872702169789375/65536*x^16 + 640266534762004953928125/8192*x^14 + 6246283669806635170340625/65536*x^12 + 1309576627087795175484375/16384*x^10 + 22786633311327636053428125/524288*x^8 + 1833407277922913245678125/131072*x^6 + 2500100833531245335015625/1048576*x^4 + 2500100833531245335015625/16777216

3, 16:

x^48 + 156*x^46 + 11295*x^44 + 503295*x^42 + 123528825/8*x^40 + 1383794685/4*x^38 + 93772703115/16*x^36 + 1227697359525/16*x^34 + 201482293787775/256*x^32 + 204035461504875/32*x^30 + 5242741241712975/128*x^28 + 26716912781033475/128*x^26 + 861774276957969375/1024*x^24 + 1366815074653153125/512*x^22 + 13523471400788724375/2048*x^20 + 25760165517575614125/2048*x^18 + 1188928924092566031375/65536*x^16 + 317195732937750541875/16384*x^14 + 974360152719430351875/65536*x^12 + 517058149136350696875/65536*x^10 + 1123264255020348065625/524288*x^8 + 267443870242940015625/262144*x^6 + 4011658053644100234375/1048576*x^4 + 2406994832186460140625/1048576*x^2 + 2406994832186460140625/16777216

4, 12:

x^48 + 96*x^46 + 4365*x^44 + 124080*x^42 + 19670805/8*x^40 + 35945910*x^38 + 6412290885/16*x^36 + 3478374900*x^34 + 6093510798375/256*x^32 + 517497455475/4*x^30 + 71722284358725/128*x^28 + 15470273043225/8*x^26 + 5430138884044875/1024*x^24 + 734384944668375/64*x^22 + 39764438367278625/2048*x^20 + 1642162530133125/64*x^18 + 1670350475948529375/65536*x^16 + 24813226487424375/2048*x^14 + 530569154345855625/65536*x^12 + 407180978916586875/4096*x^10 + 142920523599721993125/524288*x^8 + 8550800557248324375/32768*x^6 + 128262008358724865625/1048576*x^4 + 128262008358724865625/16777216

6, 8:

x^48 + 24*x^46 + 405*x^44 + 4500*x^42 + 299565/8*x^40 + 479115/2*x^38 + 19872405/16*x^36 + 5060475*x^34 + 3821253975/256*x^32 + 717099075/16*x^30 + 22544416125/128*x^28 - 3094331625/32*x^26 - 1385229124125/1024*x^24 - 1092299063625/256*x^22 + 113651706254625/2048*x^20 + 85536609109875/256*x^18 + 50047493816541375/65536*x^16 + 9681838749804375/8192*x^14 + 174648362564300625/65536*x^12 - 32197742819116875/16384*x^10 - 289779685372051875/524288*x^8 + 676152599201454375/131072*x^6 + 10142288988021815625/1048576*x^4 + 10142288988021815625/16777216

8, 6:

x^48 - 24*x^46 + 405*x^44 - 4500*x^42 + 299565/8*x^40 - 479115/2*x^38 + 19872405/16*x^36 - 5060475*x^34 + 3821253975/256*x^32 - 717099075/16*x^30 + 22544416125/128*x^28 + 3094331625/32*x^26 - 1385229124125/1024*x^24 + 1092299063625/256*x^22 + 113651706254625/2048*x^20 - 85536609109875/256*x^18 + 50047493816541375/65536*x^16 - 9681838749804375/8192*x^14 + 174648362564300625/65536*x^12 + 32197742819116875/16384*x^10 - 289779685372051875/524288*x^8 - 676152599201454375/131072*x^6 + 10142288988021815625/1048576*x^4 + 10142288988021815625/16777216

12, 4:

x^48 - 96*x^46 + 4365*x^44 - 124080*x^42 + 19670805/8*x^40 - 35945910*x^38 + 6412290885/16*x^36 - 3478374900*x^34 + 6093510798375/256*x^32 - 517497455475/4*x^30 + 71722284358725/128*x^28 - 15470273043225/8*x^26 + 5430138884044875/1024*x^24 - 734384944668375/64*x^22 + 39764438367278625/2048*x^20 - 1642162530133125/64*x^18 + 1670350475948529375/65536*x^16 - 24813226487424375/2048*x^14 + 530569154345855625/65536*x^12 - 407180978916586875/4096*x^10 + 142920523599721993125/524288*x^8 - 8550800557248324375/32768*x^6 + 128262008358724865625/1048576*x^4 + 128262008358724865625/16777216

16, 3:

x^48 - 156*x^46 + 11295*x^44 - 503295*x^42 + 123528825/8*x^40 - 1383794685/4*x^38 + 93772703115/16*x^36 - 1227697359525/16*x^34 + 201482293787775/256*x^32 - 204035461504875/32*x^30 + 5242741241712975/128*x^28 - 26716912781033475/128*x^26 + 861774276957969375/1024*x^24 - 1366815074653153125/512*x^22 + 13523471400788724375/2048*x^20 - 25760165517575614125/2048*x^18 + 1188928924092566031375/65536*x^16 - 317195732937750541875/16384*x^14 + 974360152719430351875/65536*x^12 - 517058149136350696875/65536*x^10 + 1123264255020348065625/524288*x^8 - 267443870242940015625/262144*x^6 + 4011658053644100234375/1048576*x^4 - 2406994832186460140625/1048576*x^2 + 2406994832186460140625/16777216

24, 2:

x^48 - 264*x^46 + 32085*x^44 - 2383260*x^42 + 969330285/8*x^40 - 8950625145/2*x^38 + 1990744998165/16*x^36 - 2662403938425*x^34 + 11384613336425175/256*x^32 - 9356259740976225/16*x^30 + 778081248058254525/128*x^28 - 1599697585314276525/32*x^26 + 332239535893121731875/1024*x^24 - 423123916284292348125/256*x^22 + 13417351864650579470625/2048*x^20 - 5109144223069772510625/256*x^18 + 3011093010872702169789375/65536*x^16 - 640266534762004953928125/8192*x^14 + 6246283669806635170340625/65536*x^12 - 1309576627087795175484375/16384*x^10 + 22786633311327636053428125/524288*x^8 - 1833407277922913245678125/131072*x^6 + 2500100833531245335015625/1048576*x^4 + 2500100833531245335015625/16777216

48, 1:

x^48 - 564*x^46 + 145935*x^44 - 23009085*x^42 + 19810822185/8*x^40 - 772622065215/4*x^38 + 181051103948715/16*x^36 - 8147299677692175/16*x^34 + 4570635119185310175/256*x^32 - 15743298743860512825/32*x^30 + 1369666990715864615775/128*x^28 - 23533369204118037489225/128*x^26 + 2549448330446120727999375/1024*x^24 - 13531687292367871556304375/512*x^22 + 446545680648139761358044375/2048*x^20 - 2828122644104885155267614375/2048*x^18 + 432702764548047428755944999375/65536*x^16 - 381796556954159495961127940625/16384*x^14 + 3860387409203168236940293621875/65536*x^12 - 6704883394931818516791036290625/65536*x^10 + 60343950554386366651119326615625/524288*x^8 - 20114650184795455550373108871875/262144*x^6 + 27429068433811984841417875734375/1048576*x^4 - 3577704578323302370619722921875/1048576*x^2 + 1192568192774434123539907640625/16777216

1, 49:

x^49 + 588*x^47 + 158907*x^45 + 26219655*x^43 + 23676348465/8*x^41 + 970730287065/4*x^39 + 239770380905055/16*x^37 + 11406219548769045/16*x^35 + 6786700631517581775/256*x^33 + 24884568982231133175/32*x^31 + 2314264915347495385275/128*x^29 + 42708707074140142110075/128*x^27 + 4996918727674396626878775/1024*x^25 + 28828377275044595924300625/512*x^23 + 1041939921512326109835436875/2048*x^21 + 7293579450586282768848058125/2048*x^19 + 1247202086050254353473017939375/65536*x^17 + 1247202086050254353473017939375/16384*x^15 + 14550691003919634123851875959375/65536*x^13 + 29867207850150827938432798021875/65536*x^11 + 328539286351659107322760778240625/524288*x^9 + 140802551293568188852611762103125/262144*x^7 + 268804870651357451445895182196875/1048576*x^5 + 58435841445947272053455474390625/1048576*x^3 + 58435841445947272053455474390625/16777216*x

7, 7:

x^49 + 147*x^45 + 46305/8*x^41 + 2017575/16*x^37 - 310475025/256*x^33 + 15334132275/128*x^29 - 3201395499225/1024*x^25 - 196567416478125/2048*x^21 - 6529612180100625/65536*x^17 - 147629277681125625/65536*x^13 + 8339215390151270625/524288*x^9 + 14199204583230541875/1048576*x^5 - 70996022916152709375/16777216*x

49, 1:

x^49 - 588*x^47 + 158907*x^45 - 26219655*x^43 + 23676348465/8*x^41 - 970730287065/4*x^39 + 239770380905055/16*x^37 - 11406219548769045/16*x^35 + 6786700631517581775/256*x^33 - 24884568982231133175/32*x^31 + 2314264915347495385275/128*x^29 - 42708707074140142110075/128*x^27 + 4996918727674396626878775/1024*x^25 - 28828377275044595924300625/512*x^23 + 1041939921512326109835436875/2048*x^21 - 7293579450586282768848058125/2048*x^19 + 1247202086050254353473017939375/65536*x^17 - 1247202086050254353473017939375/16384*x^15 + 14550691003919634123851875959375/65536*x^13 - 29867207850150827938432798021875/65536*x^11 + 328539286351659107322760778240625/524288*x^9 - 140802551293568188852611762103125/262144*x^7 + 268804870651357451445895182196875/1048576*x^5 - 58435841445947272053455474390625/1048576*x^3 + 58435841445947272053455474390625/16777216*x

1, 50:

x^50 + 1225/2*x^48 + 172725*x^46 + 59590125/2*x^44 + 28186129125/8*x^42 + 4853651435325/16*x^40 + 315487343296125/16*x^38 + 31683943191025125/32*x^36 + 9980442105172914375/256*x^34 + 622114224555778329375/512*x^32 + 3857108192245825642125/128*x^30 + 152531096693357650393125/256*x^28 + 9609459091681531974766875/1024*x^26 + 240236477292038299369171875/2048*x^24 + 2368045276164377522353265625/2048*x^22 + 36467897252931413844240290625/4096*x^20 + 3464450239028484315202827609375/65536*x^18 + 31180052151256358836825448484375/131072*x^16 + 51966753585427264728042414140625/65536*x^14 + 248893398751256899486939983515625/131072*x^12 + 1642696431758295536613803891203125/524288*x^10 + 3520063782339204721315294052578125/1048576*x^8 + 2240040588761312095382459851640625/1048576*x^6 + 1460896036148681801336386859765625/2097152*x^4 + 1460896036148681801336386859765625/16777216*x^2 + 58435841445947272053455474390625/33554432

2, 25:

x^50 + 575/2*x^48 + 38175*x^46 + 6218625/2*x^44 + 1392290625/8*x^42 + 113739508575/16*x^40 + 3514463827875/16*x^38 + 168097688884125/32*x^36 + 25261771038879375/256*x^34 + 752428320459440625/512*x^32 + 2232428054194713375/128*x^30 + 42304853848803170625/256*x^28 + 1278801218535560094375/1024*x^26 + 15358053668732915765625/2048*x^24 + 72806373138998733609375/2048*x^22 + 539799565539055387190625/4096*x^20 + 24713408766247444727859375/65536*x^18 + 107290652009917801941140625/131072*x^16 + 86334809439795621001171875/65536*x^14 + 199809645981425718441328125/131072*x^12 + 637763817391756250460890625/524288*x^10 + 661336196679336563619609375/1048576*x^8 + 204174901405051702359609375/1048576*x^6 + 62502520838281133375390625/2097152*x^4 + 62502520838281133375390625/16777216*x^2 - 62502520838281133375390625/33554432

5, 10:

x^50 + 125/2*x^48 + 1950*x^46 + 39375*x^44 + 4546575/8*x^42 + 98804475/16*x^40 + 104209875/2*x^38 + 1391110875/4*x^36 + 475317871875/256*x^34 + 4095400530375/512*x^32 + 1783464692625/64*x^30 + 9999427370625/128*x^28 + 181396350568125/1024*x^26 + 696920840240625/2048*x^24 + 251270779134375/512*x^22 - 24336918230625/1024*x^20 - 29812724832515625/65536*x^18 + 1339382294822446875/131072*x^16 + 1965814570078734375/32768*x^14 + 8869711533742434375/65536*x^12 + 82219236127387734375/524288*x^10 + 147994625029297921875/1048576*x^8 + 1035962375205085453125/16777216*x^2 + 207192475041017090625/33554432

10, 5:

x^50 - 125/2*x^48 + 1950*x^46 - 39375*x^44 + 4546575/8*x^42 - 98804475/16*x^40 + 104209875/2*x^38 - 1391110875/4*x^36 + 475317871875/256*x^34 - 4095400530375/512*x^32 + 1783464692625/64*x^30 - 9999427370625/128*x^28 + 181396350568125/1024*x^26 - 696920840240625/2048*x^24 + 251270779134375/512*x^22 + 24336918230625/1024*x^20 - 29812724832515625/65536*x^18 - 1339382294822446875/131072*x^16 + 1965814570078734375/32768*x^14 - 8869711533742434375/65536*x^12 + 82219236127387734375/524288*x^10 - 147994625029297921875/1048576*x^8 + 1035962375205085453125/16777216*x^2 - 207192475041017090625/33554432

25, 2:

x^50 - 575/2*x^48 + 38175*x^46 - 6218625/2*x^44 + 1392290625/8*x^42 - 113739508575/16*x^40 + 3514463827875/16*x^38 - 168097688884125/32*x^36 + 25261771038879375/256*x^34 - 752428320459440625/512*x^32 + 2232428054194713375/128*x^30 - 42304853848803170625/256*x^28 + 1278801218535560094375/1024*x^26 - 15358053668732915765625/2048*x^24 + 72806373138998733609375/2048*x^22 - 539799565539055387190625/4096*x^20 + 24713408766247444727859375/65536*x^18 - 107290652009917801941140625/131072*x^16 + 86334809439795621001171875/65536*x^14 - 199809645981425718441328125/131072*x^12 + 637763817391756250460890625/524288*x^10 - 661336196679336563619609375/1048576*x^8 + 204174901405051702359609375/1048576*x^6 - 62502520838281133375390625/2097152*x^4 + 62502520838281133375390625/16777216*x^2 + 62502520838281133375390625/33554432

50, 1:

x^50 - 1225/2*x^48 + 172725*x^46 - 59590125/2*x^44 + 28186129125/8*x^42 - 4853651435325/16*x^40 + 315487343296125/16*x^38 - 31683943191025125/32*x^36 + 9980442105172914375/256*x^34 - 622114224555778329375/512*x^32 + 3857108192245825642125/128*x^30 - 152531096693357650393125/256*x^28 + 9609459091681531974766875/1024*x^26 - 240236477292038299369171875/2048*x^24 + 2368045276164377522353265625/2048*x^22 - 36467897252931413844240290625/4096*x^20 + 3464450239028484315202827609375/65536*x^18 - 31180052151256358836825448484375/131072*x^16 + 51966753585427264728042414140625/65536*x^14 - 248893398751256899486939983515625/131072*x^12 + 1642696431758295536613803891203125/524288*x^10 - 3520063782339204721315294052578125/1048576*x^8 + 2240040588761312095382459851640625/1048576*x^6 - 1460896036148681801336386859765625/2097152*x^4 + 1460896036148681801336386859765625/16777216*x^2 - 58435841445947272053455474390625/33554432

Definition

The monic generalized Hermite polynomial for Painlevé IV is $\widehat H_{m,n}(x)=W_x(H_m,H_{m+1},\ldots,H_{m+n-1})/\bigl(2^{mn+n(n-1)/2}\prod_{j=1}^{n-1}j!\bigr)$ [2]. Here $W_x(f_1,\ldots,f_n)=\det(\partial_x^{i-1}f_j)_{i,j=1}^n$, and $H_k(x)$ is defined by $\exp(2xt-t^2)=\sum_{k\geq0}H_k(x)t^k/k!$.

Parameters

$m$

— first Hermite degree ($m\geq1$)

$n$

— number of Wronskian factors ($n\geq1$)

Formulas

(1)

$\deg\widehat H_{m,n}=mn$, and $\widehat H_{m,n}(-x)=(-1)^{mn}\widehat H_{m,n}(x)$.

(2)

$\widehat H_{m,1}(x)=2^{-m}H_m(x)$ and $\widehat H_{1,n}(x)=2^{-n}i^{-n}H_n(ix)$, where $H_k$ is the physicists' Hermite polynomial and $i^2=-1$ [1], equation (5).

(3)

$\widehat H_{n,m}(x)=i^{-mn}\widehat H_{m,n}(ix)$, where $i^2=-1$ [1], equation (7).

(4)

Writing $P=\widehat H_{m,n}(x)$ and taking derivatives in $x$, $\widehat H_{m+1,n}\widehat H_{m-1,n}=P^2+(PP^{\prime\prime}-(P^\prime)^2)/(2m)$ and $\widehat H_{m,n+1}\widehat H_{m,n-1}=P^2-(PP^{\prime\prime}-(P^\prime)^2)/(2n)$ for $m,n\geq1$ [1], equation (3).

(5)

For $m,n\geq1$, let $w=\widehat H_{m+1,n}^{\prime}/\widehat H_{m+1,n}-\widehat H_{m,n}^{\prime}/\widehat H_{m,n}$. Then $w^{\prime\prime}=(w^\prime)^2/(2w)+3w^3/2+4xw^2+2(x^2-\alpha)w+\beta/w$, with $\alpha=2m+n+1$ and $\beta=-2n^2$, as a rational-function identity [1], equations (1), (2), and (8).

Comments

(6)

The hat denotes monic normalization in the unchanged variable $x$. The Wronskian uses the physicists' Hermite polynomials, whose variable scaling differs from the probabilists' convention.

(7)

The table lists positive indices. For the boundary terms in the recurrences, set $\widehat H_{0,n}(x)=\widehat H_{m,0}(x)=1$, with the empty Wronskian equal to $1$.

(8)

The generalized Hermite normalization in [1] is $H_{m,n}(x)=2^{mn}\widehat H_{m,n}(x)$. Multiplicative normalization does not affect the logarithmic derivative in (5).

Programs

(P1)

Sage

# Compute one monic generalized Hermite polynomial in SageMath.
from math import factorial
R = PolynomialRing(QQ, 'x')
x = R.gen()
def generalized_hermite(m, n):
    if m == 0 or n == 0:
        return R.one()
    previous = R.one()
    current = sum((QQ((-1)**j * factorial(m)) /
                   QQ(4**j * factorial(j) * factorial(m-2*j))) * x**(m-2*j)
                  for j in range(m//2+1))
    for k in range(1, n):
        numerator = current**2 + (current.derivative()**2 -
                    current * current.derivative(2)) / QQ(2*k)
        following, remainder = numerator.quo_rem(previous)
        assert remainder == 0
        previous, current = current, following
    return current
generalized_hermite(2, 2)  # x^4 + 3/4

References

[1]

Robert Buckingham, Large-Degree Asymptotics of Rational Painlevé-IV Functions Associated to Generalized Hermite Polynomials, International Mathematics Research Notices 2020 (18), 5534–5577. (doi)

Links

[2]

P. A. Clarkson, The Fourth Painlevé Transcendent (2008), Section 5.1, especially equation (5.8)

Similar tables

Hermite polynomials in physicist's convention — supply the consecutive factors of the defining Wronskian and the n=1 specialization

Hermite polynomials in probabilist's convention — give the classical Hermite family with a different variable scaling

Data properties

Entries are of type: rational polynomial

How well the digits are known: exact

Table is complete: no (it holds every positive pair $(m,n)$ with $mn\leq50$, covering all Wronskian shapes through degree $50$, including both orientations, the classical $n=1$ slice, and balanced cases through $(7,7)$. This noncanonical degree bound is a practical readability choice; the longest expanded polynomial has $920$ characters. The identically unit boundary axes are stated in the comments)

How they were obtained:

Computed over $\mathbb{Q}[x]$ from the physicists' Hermite recurrence and the defining Wronskian, using fraction-free elimination with zero-remainder tests and exact leading-coefficient normalization. All $207$ entries were independently reconstructed from a closed Hermite coefficient sum and the differential recurrence in $n$.

Both differential recurrences, degree, monicity, parity, transpose symmetry, the fourth-order bilinear equation in [2], equation (5.9), and the rational Painlevé IV identity were checked exactly on every entry, using neighboring polynomials when required. Known and deliberately perturbed controls were tested. The same independent checks were repeated on the stored values, and the attached generator verified every stored entry.