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Krawtchouk polynomials $\mathcal{K}_k(x)$ of the Hamming scheme
edit · history · discussion · files · long url · Sheffer sequence combinatorics polynomial orthogonal polynomials
Polynomials
$q$
$N$
$k$ 
$\mathcal{K}_k(x;N,q)$
2
1
0:
1
2
1
1:
-2*x + 1
2
2
0:
1
2
2
1:
-2*x + 2
2
2
2:
2*x^2 - 4*x + 1
2
3
0:
1
2
3
1:
-2*x + 3
2
3
2:
2*x^2 - 6*x + 3
2
3
3:
-4/3*x^3 + 6*x^2 - 20/3*x + 1
2
4
0:
1
2
4
1:
-2*x + 4
2
4
2:
2*x^2 - 8*x + 6
2
4
3:
-4/3*x^3 + 8*x^2 - 38/3*x + 4
2
4
4:
2/3*x^4 - 16/3*x^3 + 40/3*x^2 - 32/3*x + 1
2
5
0:
1
2
5
1:
-2*x + 5
2
5
2:
2*x^2 - 10*x + 10
2
5
3:
-4/3*x^3 + 10*x^2 - 62/3*x + 10
2
5
4:
2/3*x^4 - 20/3*x^3 + 64/3*x^2 - 70/3*x + 5
2
5
5:
-4/15*x^5 + 10/3*x^4 - 44/3*x^3 + 80/3*x^2 - 256/15*x + 1
2
6
0:
1
2
6
1:
-2*x + 6
2
6
2:
2*x^2 - 12*x + 15
2
6
3:
-4/3*x^3 + 12*x^2 - 92/3*x + 20
2
6
4:
2/3*x^4 - 8*x^3 + 94/3*x^2 - 44*x + 15
2
6
5:
-4/15*x^5 + 4*x^4 - 64/3*x^3 + 48*x^2 - 202/5*x + 6
2
6
6:
4/45*x^6 - 8/5*x^5 + 98/9*x^4 - 104/3*x^3 + 2296/45*x^2 - 416/15*x + 1
2
7
0:
1
2
7
1:
-2*x + 7
2
7
2:
2*x^2 - 14*x + 21
2
7
3:
-4/3*x^3 + 14*x^2 - 128/3*x + 35
2
7
4:
2/3*x^4 - 28/3*x^3 + 130/3*x^2 - 224/3*x + 35
2
7
5:
-4/15*x^5 + 14/3*x^4 - 88/3*x^3 + 238/3*x^2 - 422/5*x + 21
2
7
6:
4/45*x^6 - 28/15*x^5 + 134/9*x^4 - 56*x^3 + 4456/45*x^2 - 1022/15*x + 7
2
7
7:
-8/315*x^7 + 28/45*x^6 - 272/45*x^5 + 266/9*x^4 - 3416/45*x^3 + 4312/45*x^2 - 4832/105*x + 1
2
8
0:
1
2
8
1:
-2*x + 8
2
8
2:
2*x^2 - 16*x + 28
2
8
3:
-4/3*x^3 + 16*x^2 - 170/3*x + 56
2
8
4:
2/3*x^4 - 32/3*x^3 + 172/3*x^2 - 352/3*x + 70
2
8
5:
-4/15*x^5 + 16/3*x^4 - 116/3*x^3 + 368/3*x^2 - 2386/15*x + 56
2
8
6:
4/45*x^6 - 32/15*x^5 + 176/9*x^4 - 256/3*x^3 + 8026/45*x^2 - 2288/15*x + 28
2
8
7:
-8/315*x^7 + 32/45*x^6 - 356/45*x^5 + 400/9*x^4 - 5936/45*x^3 + 8768/45*x^2 - 11986/105*x + 8
2
8
8:
2/315*x^8 - 64/315*x^7 + 8/3*x^6 - 832/45*x^5 + 1088/15*x^4 - 7168/45*x^3 + 11264/63*x^2 - 8192/105*x + 1
2
9
0:
1
2
9
1:
-2*x + 9
2
9
2:
2*x^2 - 18*x + 36
2
9
3:
-4/3*x^3 + 18*x^2 - 218/3*x + 84
2
9
4:
2/3*x^4 - 12*x^3 + 220/3*x^2 - 174*x + 126
2
9
5:
-4/15*x^5 + 6*x^4 - 148/3*x^3 + 180*x^2 - 1382/5*x + 126
2
9
6:
4/45*x^6 - 12/5*x^5 + 224/9*x^4 - 124*x^3 + 13546/45*x^2 - 1558/5*x + 84
2
9
7:
-8/315*x^7 + 4/5*x^6 - 452/45*x^5 + 64*x^4 - 9776/45*x^3 + 1866/5*x^2 - 9334/35*x + 36
2
9
8:
2/315*x^8 - 8/35*x^7 + 152/45*x^6 - 132/5*x^5 + 5264/45*x^4 - 1456/5*x^3 + 39232/105*x^2 - 6726/35*x + 9
2
9
9:
-4/2835*x^9 + 2/35*x^8 - 184/189*x^7 + 136/15*x^6 - 6772/135*x^5 + 2512/15*x^4 - 184816/567*x^3 + 35008/105*x^2 - 42496/315*x + 1
2
10
0:
1
2
10
1:
-2*x + 10
2
10
2:
2*x^2 - 20*x + 45
2
10
3:
-4/3*x^3 + 20*x^2 - 272/3*x + 120
2
10
4:
2/3*x^4 - 40/3*x^3 + 274/3*x^2 - 740/3*x + 210
2
10
5:
-4/15*x^5 + 20/3*x^4 - 184/3*x^3 + 760/3*x^2 - 2252/5*x + 252
2
10
6:
4/45*x^6 - 8/3*x^5 + 278/9*x^4 - 520/3*x^3 + 21646/45*x^2 - 588*x + 210
2
10
7:
-8/315*x^7 + 8/9*x^6 - 112/9*x^5 + 800/9*x^4 - 15356/45*x^3 + 6068/9*x^2 - 4048/7*x + 120
2
10
8:
2/315*x^8 - 16/63*x^7 + 188/45*x^6 - 328/9*x^5 + 8144/45*x^4 - 4576/9*x^3 + 78418/105*x^2 - 3212/7*x + 45
2
10
9:
-4/2835*x^9 + 4/63*x^8 - 1136/945*x^7 + 112/9*x^6 - 10336/135*x^5 + 2560/9*x^4 - 1749632/2835*x^3 + 14848/21*x^2 - 20606/63*x + 10
2
10
10:
4/14175*x^10 - 8/567*x^9 + 286/945*x^8 - 688/189*x^7 + 18172/675*x^6 - 17032/135*x^5 + 1053712/2835*x^4 - 372448/567*x^3 + 981376/1575*x^2 - 74752/315*x + 1
2
11
0:
1
2
11
1:
-2*x + 11
2
11
2:
2*x^2 - 22*x + 55
2
11
3:
-4/3*x^3 + 22*x^2 - 332/3*x + 165
2
11
4:
2/3*x^4 - 44/3*x^3 + 334/3*x^2 - 1012/3*x + 330
2
11
5:
-4/15*x^5 + 22/3*x^4 - 224/3*x^3 + 1034/3*x^2 - 10456/15*x + 462
2
11
6:
4/45*x^6 - 44/15*x^5 + 338/9*x^4 - 704/3*x^3 + 33046/45*x^2 - 5192/5*x + 462
2
11
7:
-8/315*x^7 + 44/45*x^6 - 136/9*x^5 + 1078/9*x^4 - 23156/45*x^3 + 51986/45*x^2 - 8164/7*x + 330
2
11
8:
2/315*x^8 - 88/315*x^7 + 76/15*x^6 - 440/9*x^5 + 4048/15*x^4 - 38236/45*x^3 + 447634/315*x^2 - 7260/7*x + 165
2
11
9:
-4/2835*x^9 + 22/315*x^8 - 1376/945*x^7 + 748/45*x^6 - 15256/135*x^5 + 20944/45*x^4 - 3191072/2835*x^3 + 50886/35*x^2 - 49514/63*x + 55
2
11
10:
4/14175*x^10 - 44/2835*x^9 + 346/945*x^8 - 4576/945*x^7 + 26572/675*x^6 - 27368/135*x^5 + 1860112/2835*x^4 - 3611872/2835*x^3 + 2094976/1575*x^2 - 177782/315*x + 11
2
11
11:
-8/155925*x^11 + 44/14175*x^10 - 232/2835*x^9 + 1166/945*x^8 - 55448/4725*x^7 + 49412/675*x^6 - 170936/567*x^5 + 2276912/2835*x^4 - 18586528/14175*x^3 + 1840256/1575*x^2 - 1467392/3465*x + 1
2
12
0:
1
2
12
1:
-2*x + 12
2
12
2:
2*x^2 - 24*x + 66
2
12
3:
-4/3*x^3 + 24*x^2 - 398/3*x + 220
2
12
4:
2/3*x^4 - 16*x^3 + 400/3*x^2 - 448*x + 495
2
12
5:
-4/15*x^5 + 8*x^4 - 268/3*x^3 + 456*x^2 - 5172/5*x + 792
2
12
6:
4/45*x^6 - 16/5*x^5 + 404/9*x^4 - 928/3*x^3 + 48556/45*x^2 - 26032/15*x + 924
2
12
7:
-8/315*x^7 + 16/15*x^6 - 812/45*x^5 + 472/3*x^4 - 33716/45*x^3 + 9448/5*x^2 - 77164/35*x + 792
2
12
8:
2/315*x^8 - 32/105*x^7 + 272/45*x^6 - 64*x^5 + 17534/45*x^4 - 20464/15*x^3 + 270512/105*x^2 - 15424/7*x + 495
2
12
9:
-4/2835*x^9 + 8/105*x^8 - 328/189*x^7 + 976/45*x^6 - 21856/135*x^5 + 33088/45*x^4 - 1119988/567*x^3 + 905608/315*x^2 - 114854/63*x + 220
2
12
10:
4/14175*x^10 - 16/945*x^9 + 412/945*x^8 - 1984/315*x^7 + 37792/675*x^6 - 4736/15*x^5 + 3179584/2835*x^4 - 2267648/945*x^3 + 4384846/1575*x^2 - 141784/105*x + 66
2
12
11:
-8/155925*x^11 + 16/4725*x^10 - 92/945*x^9 + 8/5*x^8 - 78328/4725*x^7 + 25328/225*x^6 - 1429408/2835*x^5 + 1379008/945*x^4 - 12215296/4725*x^3 + 62464/25*x^2 - 1140998/1155*x + 12
2
12
12:
4/467775*x^12 - 32/51975*x^11 + 832/42525*x^10 - 1024/2835*x^9 + 60814/14175*x^8 - 162016/4725*x^7 + 7963696/42525*x^6 - 656576/945*x^5 + 72429344/42525*x^4 - 36879616/14175*x^3 + 114303488/51975*x^2 - 2650112/3465*x + 1
2
13
0:
1
2
13
1:
-2*x + 13
2
13
2:
2*x^2 - 26*x + 78
2
13
3:
-4/3*x^3 + 26*x^2 - 470/3*x + 286
2
13
4:
2/3*x^4 - 52/3*x^3 + 472/3*x^2 - 1742/3*x + 715
2
13
5:
-4/15*x^5 + 26/3*x^4 - 316/3*x^3 + 1768/3*x^2 - 7412/5*x + 1287
2
13
6:
4/45*x^6 - 52/15*x^5 + 476/9*x^4 - 1196/3*x^3 + 69076/45*x^2 - 41548/15*x + 1716
2
13
7:
-8/315*x^7 + 52/45*x^6 - 956/45*x^5 + 1820/9*x^4 - 47636/45*x^3 + 133588/45*x^2 - 413716/105*x + 1716
2
13
8:
2/315*x^8 - 104/315*x^7 + 64/9*x^6 - 3692/45*x^5 + 24614/45*x^4 - 95108/45*x^3 + 93784/21*x^2 - 154284/35*x + 1287
2
13
9:
-4/2835*x^9 + 26/315*x^8 - 1928/945*x^7 + 416/15*x^6 - 30496/135*x^5 + 16874/15*x^4 - 9467636/2835*x^3 + 1717144/315*x^2 - 253670/63*x + 715
2
13
10:
4/14175*x^10 - 52/2835*x^9 + 484/945*x^8 - 7592/945*x^7 + 52432/675*x^6 - 12896/27*x^5 + 5264128/2835*x^4 - 12402884/2835*x^3 + 2970962/525*x^2 - 999622/315*x + 286
2
13
11:
-8/155925*x^11 + 52/14175*x^10 - 4/35*x^9 + 1924/945*x^8 - 108088/4725*x^7 + 113776/675*x^6 - 2324512/2835*x^5 + 7316608/2835*x^4 - 23553536/4725*x^3 + 8320078/1575*x^2 - 2700622/1155*x + 78
2
13
12:
4/467775*x^12 - 104/155925*x^11 + 976/42525*x^10 - 260/567*x^9 + 83494/14175*x^8 - 240344/4725*x^7 + 12750688/42525*x^6 - 3399136/2835*x^5 + 134484704/42525*x^4 - 73525504/14175*x^3 + 244166144/51975*x^2 - 6073106/3465*x + 13
2
13
13:
-8/6081075*x^13 + 52/467775*x^12 - 56/13365*x^11 + 3952/42525*x^10 - 2728/2025*x^9 + 189982/14175*x^8 - 795016/8505*x^7 + 19492096/42525*x^6 - 9473248/6075*x^5 + 151535072/42525*x^4 - 22911232/4455*x^3 + 215940608/51975*x^2 - 62836736/45045*x + 1
2
14
0:
1
2
14
1:
-2*x + 14
2
14
2:
2*x^2 - 28*x + 91
2
14
3:
-4/3*x^3 + 28*x^2 - 548/3*x + 364
2
14
4:
2/3*x^4 - 56/3*x^3 + 550/3*x^2 - 2212/3*x + 1001
2
14
5:
-4/15*x^5 + 28/3*x^4 - 368/3*x^3 + 2240/3*x^2 - 30946/15*x + 2002
2
14
6:
4/45*x^6 - 56/15*x^5 + 554/9*x^4 - 504*x^3 + 95596/45*x^2 - 63784/15*x + 3003
2
14
7:
-8/315*x^7 + 56/45*x^6 - 1112/45*x^5 + 2296/9*x^4 - 65576/45*x^3 + 202664/45*x^2 - 704552/105*x + 3432
2
14
8:
2/315*x^8 - 16/45*x^7 + 124/15*x^6 - 4648/45*x^5 + 3746/5*x^4 - 142744/45*x^3 + 2341876/315*x^2 - 125224/15*x + 3003
2
14
9:
-4/2835*x^9 + 4/45*x^8 - 64/27*x^7 + 1568/45*x^6 - 41572/135*x^5 + 75236/45*x^4 - 3091888/567*x^3 + 446272/45*x^2 - 379558/45*x + 2002
2
14
10:
4/14175*x^10 - 8/405*x^9 + 562/945*x^8 - 272/27*x^7 + 71152/675*x^6 - 94976/135*x^5 + 8453314/2835*x^4 - 624872/81*x^3 + 17498606/1575*x^2 - 323996/45*x + 1001
2
14
11:
-8/155925*x^11 + 8/2025*x^10 - 376/2835*x^9 + 344/135*x^8 - 20864/675*x^7 + 166208/675*x^6 - 3678592/2835*x^5 + 1797248/405*x^4 - 132675028/14175*x^3 + 2461852/225*x^2 - 2728244/495*x + 364
2
14
12:
4/467775*x^12 - 16/22275*x^11 + 1132/42525*x^10 - 232/405*x^9 + 112354/14175*x^8 - 16592/225*x^7 + 19918576/42525*x^6 - 817664/405*x^5 + 244233824/42525*x^4 - 20598016/2025*x^3 + 518728718/51975*x^2 - 2024996/495*x + 91
2
14
13:
-8/6081075*x^13 + 8/66825*x^12 - 2272/467775*x^11 + 704/6075*x^10 - 316/175*x^9 + 39068/2025*x^8 - 6138176/42525*x^7 + 4606112/6075*x^6 - 117299776/42525*x^5 + 40859968/6075*x^4 - 1610673664/155925*x^3 + 65729536/7425*x^2 - 20255302/6435*x + 14
2
14
14:
8/42567525*x^14 - 16/868725*x^13 + 76/93555*x^12 - 1424/66825*x^11 + 2248/6075*x^10 - 112/25*x^9 + 2308822/59535*x^8 - 10296304/42525*x^7 + 46154384/42525*x^6 - 20842112/6075*x^5 + 98648864/13365*x^4 - 225897728/22275*x^3 + 37253047808/4729725*x^2 - 115552256/45045*x + 1
2
15
0:
1
2
15
1:
-2*x + 15
2
15
2:
2*x^2 - 30*x + 105
2
15
3:
-4/3*x^3 + 30*x^2 - 632/3*x + 455
2
15
4:
2/3*x^4 - 20*x^3 + 634/3*x^2 - 920*x + 1365
2
15
5:
-4/15*x^5 + 10*x^4 - 424/3*x^3 + 930*x^2 - 14002/5*x + 3003
2
15
6:
4/45*x^6 - 4*x^5 + 638/9*x^4 - 1880/3*x^3 + 129196/45*x^2 - 18946/3*x + 5005
2
15
7:
-8/315*x^7 + 4/3*x^6 - 256/9*x^5 + 950/3*x^4 - 88256/45*x^3 + 6628*x^2 - 76736/7*x + 6435
2
15
8:
2/315*x^8 - 8/21*x^7 + 428/45*x^6 - 128*x^5 + 45194/45*x^4 - 13888/3*x^3 + 417836/35*x^2 - 105408/7*x + 6435
2
15
9:
-4/2835*x^9 + 2/21*x^8 - 368/135*x^7 + 388/9*x^6 - 55516/135*x^5 + 21790/9*x^4 - 24452312/2835*x^3 + 1093156/63*x^2 - 151046/9*x + 5005
2
15
10:
4/14175*x^10 - 4/189*x^9 + 646/945*x^8 - 112/9*x^7 + 94672/675*x^6 - 15172/15*x^5 + 13193182/2835*x^4 - 2488664/189*x^3 + 33118126/1575*x^2 - 234518/15*x + 3003
2
15
11:
-8/155925*x^11 + 4/945*x^10 - 16/105*x^9 + 22/7*x^8 - 27664/675*x^7 + 15824/45*x^6 - 5673088/2835*x^5 + 1402270/189*x^4 - 80675876/4725*x^3 + 2315438/105*x^2 - 419480/33*x + 1365
2
15
12:
4/467775*x^12 - 8/10395*x^11 + 52/1701*x^10 - 400/567*x^9 + 148474/14175*x^8 - 14128/135*x^7 + 6077936/8505*x^6 - 626816/189*x^5 + 432944864/42525*x^4 - 55372228/2835*x^3 + 217483306/10395*x^2 - 950648/99*x + 455
2
15
13:
-8/6081075*x^13 + 4/31185*x^12 - 2608/467775*x^11 + 404/2835*x^10 - 33716/14175*x^9 + 2858/105*x^8 - 9274064/42525*x^7 + 3477424/2835*x^6 - 203154496/42525*x^5 + 7070048/567*x^4 - 1065573632/51975*x^3 + 65255698/3465*x^2 - 3105350/429*x + 105
2
15
14:
8/42567525*x^14 - 8/405405*x^13 + 436/467775*x^12 - 272/10395*x^11 + 328/675*x^10 - 44/7*x^9 + 17287106/297675*x^8 - 1095632/2835*x^7 + 78397168/42525*x^6 - 5848768/945*x^5 + 314234656/22275*x^4 - 212797184/10395*x^3 + 5274850816/315315*x^2 - 17155958/3003*x + 15
2
15
15:
-16/638512875*x^15 + 8/2837835*x^14 - 2624/18243225*x^13 + 412/93555*x^12 - 632992/7016625*x^11 + 7928/6075*x^10 - 12244928/893025*x^9 + 701014/6615*x^8 - 384763408/637875*x^7 + 106534928/42525*x^6 - 10431111808/1403325*x^5 + 40560032/2673*x^4 - 1416651904256/70945875*x^3 + 70848976384/4729725*x^2 - 42790912/9009*x + 1
2
16
0:
1
2
16
1:
-2*x + 16
2
16
2:
2*x^2 - 32*x + 120
2
16
3:
-4/3*x^3 + 32*x^2 - 722/3*x + 560
2
16
4:
2/3*x^4 - 64/3*x^3 + 724/3*x^2 - 3392/3*x + 1820
2
16
5:
-4/15*x^5 + 32/3*x^4 - 484/3*x^3 + 3424/3*x^2 - 18602/5*x + 4368
2
16
6:
4/45*x^6 - 64/15*x^5 + 728/9*x^4 - 768*x^3 + 171046/45*x^2 - 136736/15*x + 8008
2
16
7:
-8/315*x^7 + 64/45*x^6 - 292/9*x^5 + 3488/9*x^4 - 116456/45*x^3 + 427456/45*x^2 - 362830/21*x + 11440
2
16
8:
2/315*x^8 - 128/315*x^7 + 488/45*x^6 - 1408/9*x^5 + 59444/45*x^4 - 296576/45*x^3 + 649816/35*x^2 - 182144/7*x + 12870
2
16
9:
-4/2835*x^9 + 32/315*x^8 - 2936/945*x^7 + 2368/45*x^6 - 72796/135*x^5 + 154144/45*x^4 - 37576472/2835*x^3 + 9226304/315*x^2 - 2005994/63*x + 11440
2
16
10:
4/14175*x^10 - 64/2835*x^9 + 736/945*x^8 - 2048/135*x^7 + 123772/675*x^6 - 192064/135*x^5 + 20057032/2835*x^4 - 61782272/2835*x^3 + 60447026/1575*x^2 - 1458784/45*x + 8008
2
16
11:
-8/155925*x^11 + 64/14175*x^10 - 164/945*x^9 + 3616/945*x^8 - 36064/675*x^7 + 332032/675*x^6 - 8540596/2835*x^5 + 34227232/2835*x^4 - 142892476/4725*x^3 + 67849696/1575*x^2 - 4677098/165*x + 4368
2
16
12:
4/467775*x^12 - 128/155925*x^11 + 296/8505*x^10 - 2432/2835*x^9 + 193024/14175*x^8 - 32768/225*x^7 + 9068672/8505*x^6 - 15075328/2835*x^5 + 748455614/42525*x^4 - 518888768/14175*x^3 + 446711668/10395*x^2 - 2209088/99*x + 1820
2
16
13:
-8/6081075*x^13 + 64/467775*x^12 - 424/66825*x^11 + 1472/8505*x^10 - 14572/4725*x^9 + 534304/14175*x^8 - 13724384/42525*x^7 + 16510208/8505*x^6 - 49169728/6075*x^5 + 963198464/42525*x^4 - 6242193436/155925*x^3 + 82650080/2079*x^2 - 21674474/1287*x + 560
2
16
14:
8/42567525*x^14 - 128/6081075*x^13 + 496/467775*x^12 - 14848/467775*x^11 + 8908/14175*x^10 - 122816/14175*x^9 + 25389536/297675*x^8 - 25708544/42525*x^7 + 130558528/42525*x^6 - 466349056/42525*x^5 + 4143905792/155925*x^4 - 78872576/1925*x^3 + 11213119334/315315*x^2 - 38893408/3003*x + 120
2
16
15:
-16/638512875*x^15 + 128/42567525*x^14 - 2984/18243225*x^13 + 832/155925*x^12 - 116656/1002375*x^11 + 2176/1215*x^10 - 17858228/893025*x^9 + 48832736/297675*x^8 - 90182944/91125*x^7 + 61644032/14175*x^6 - 2730933184/200475*x^5 + 1956704768/66825*x^4 - 2868992685056/70945875*x^3 + 149971738624/4729725*x^2 - 94258786/9009*x + 16
2
16
16:
2/638512875*x^16 - 256/638512875*x^15 + 272/11609325*x^14 - 15104/18243225*x^13 + 138584/7016625*x^12 - 2359552/7016625*x^11 + 3761488/893025*x^10 - 35203328/893025*x^9 + 1237256192/4465125*x^8 - 929824768/637875*x^7 + 7953771776/1403325*x^6 - 22329192448/1403325*x^5 + 600083910656/19348875*x^4 - 2786833596416/70945875*x^3 + 135132872704/4729725*x^2 - 79691776/9009*x + 1
3
1
0:
1
3
1
1:
-3*x + 2
3
2
0:
1
3
2
1:
-3*x + 4
3
2
2:
9/2*x^2 - 21/2*x + 4
3
3
0:
1
3
3
1:
-3*x + 6
3
3
2:
9/2*x^2 - 33/2*x + 12
3
3
3:
-9/2*x^3 + 45/2*x^2 - 30*x + 8
3
4
0:
1
3
4
1:
-3*x + 8
3
4
2:
9/2*x^2 - 45/2*x + 24
3
4
3:
-9/2*x^3 + 63/2*x^2 - 63*x + 32
3
4
4:
27/8*x^4 - 117/4*x^3 + 657/8*x^2 - 321/4*x + 16
3
5
0:
1
3
5
1:
-3*x + 10
3
5
2:
9/2*x^2 - 57/2*x + 40
3
5
3:
-9/2*x^3 + 81/2*x^2 - 108*x + 80
3
5
4:
27/8*x^4 - 153/4*x^3 + 1161/8*x^2 - 825/4*x + 80
3
5
5:
-81/40*x^5 + 27*x^4 - 1035/8*x^3 + 531/2*x^2 - 2091/10*x + 32
3
6
0:
1
3
6
1:
-3*x + 12
3
6
2:
9/2*x^2 - 69/2*x + 60
3
6
3:
-9/2*x^3 + 99/2*x^2 - 165*x + 160
3
6
4:
27/8*x^4 - 189/4*x^3 + 1809/8*x^2 - 1689/4*x + 240
3
6
5:
-81/40*x^5 + 135/4*x^4 - 1647/8*x^3 + 2223/4*x^2 - 3108/5*x + 192
3
6
6:
81/80*x^6 - 1539/80*x^5 + 2241/16*x^4 - 7785/16*x^3 + 32337/40*x^2 - 5397/10*x + 64
3
7
0:
1
3
7
1:
-3*x + 14
3
7
2:
9/2*x^2 - 81/2*x + 84
3
7
3:
-9/2*x^3 + 117/2*x^2 - 234*x + 280
3
7
4:
27/8*x^4 - 225/4*x^3 + 2601/8*x^2 - 3009/4*x + 560
3
7
5:
-81/40*x^5 + 81/2*x^4 - 2403/8*x^3 + 1008*x^2 - 14661/10*x + 672
3
7
6:
81/80*x^6 - 1863/80*x^5 + 3321/16*x^4 - 14373/16*x^3 + 76797/40*x^2 - 17829/10*x + 448
3
7
7:
-243/560*x^7 + 891/80*x^6 - 9153/80*x^5 + 9585/16*x^4 - 67113/40*x^3 + 23823/10*x^2 - 48714/35*x + 128
3
8
0:
1
3
8
1:
-3*x + 16
3
8
2:
9/2*x^2 - 93/2*x + 112
3
8
3:
-9/2*x^3 + 135/2*x^2 - 315*x + 448
3
8
4:
27/8*x^4 - 261/4*x^3 + 3537/8*x^2 - 4881/4*x + 1120
3
8
5:
-81/40*x^5 + 189/4*x^4 - 3303/8*x^3 + 6633/4*x^2 - 14853/5*x + 1792
3
8
6:
81/80*x^6 - 2187/80*x^5 + 4617/16*x^4 - 23985/16*x^3 + 157437/40*x^2 - 47151/10*x + 1792
3
8
7:
-243/560*x^7 + 1053/80*x^6 - 12879/80*x^5 + 16227/16*x^4 - 69489/20*x^3 + 124443/20*x^2 - 173517/35*x + 1024
3
8
8:
729/4480*x^8 - 1215/224*x^7 + 4779/64*x^6 - 43821/80*x^5 + 1471743/640*x^4 - 175761/32*x^3 + 1543599/224*x^2 - 1009059/280*x + 256
3
9
0:
1
3
9
1:
-3*x + 18
3
9
2:
9/2*x^2 - 105/2*x + 144
3
9
3:
-9/2*x^3 + 153/2*x^2 - 408*x + 672
3
9
4:
27/8*x^4 - 297/4*x^3 + 4617/8*x^2 - 7401/4*x + 2016
3
9
5:
-81/40*x^5 + 54*x^4 - 4347/8*x^3 + 5085/2*x^2 - 54111/10*x + 4032
3
9
6:
81/80*x^6 - 2511/80*x^5 + 6129/16*x^4 - 37197/16*x^3 + 290097/40*x^2 - 106563/10*x + 5376
3
9
7:
-243/560*x^7 + 243/16*x^6 - 17253/80*x^5 + 25461/16*x^4 - 258903/40*x^3 + 14094*x^2 - 503574/35*x + 4608
3
9
8:
729/4480*x^8 - 7047/1120*x^7 + 32319/320*x^6 - 69579/80*x^5 + 2769903/640*x^4 - 1990629/160*x^3 + 21655611/1120*x^2 - 3785331/280*x + 2304
3
9
9:
-243/4480*x^9 + 729/320*x^8 - 90639/2240*x^7 + 63261/160*x^6 - 1480437/640*x^5 + 2639601/320*x^4 - 19479303/1120*x^3 + 1578087/80*x^2 - 1315239/140*x + 512
3
10
0:
1
3
10
1:
-3*x + 20
3
10
2:
9/2*x^2 - 117/2*x + 180
3
10
3:
-9/2*x^3 + 171/2*x^2 - 513*x + 960
3
10
4:
27/8*x^4 - 333/4*x^3 + 5841/8*x^2 - 10665/4*x + 3360
3
10
5:
-81/40*x^5 + 243/4*x^4 - 5535/8*x^3 + 14787/4*x^2 - 45558/5*x + 8064
3
10
6:
81/80*x^6 - 567/16*x^5 + 7857/16*x^4 - 54585/16*x^3 + 493497/40*x^2 - 42957/2*x + 13440
3
10
7:
-243/560*x^7 + 1377/80*x^6 - 4455/16*x^5 + 37719/16*x^4 - 55611/5*x^3 + 571977/20*x^2 - 249903/7*x + 15360
3
10
8:
729/4480*x^8 - 8019/1120*x^7 + 42039/320*x^6 - 20817/16*x^5 + 4806783/640*x^4 - 4061853/160*x^3 + 53226171/1120*x^2 - 2368503/56*x + 11520
3
10
9:
-243/4480*x^9 + 729/280*x^8 - 118827/2240*x^7 + 4779/8*x^6 - 2593701/640*x^5 + 169047/10*x^4 - 47348109/1120*x^3 + 1635111/28*x^2 - 510057/14*x + 5120
3
10
10:
729/44800*x^10 - 7533/8960*x^9 + 16767/896*x^8 - 1051461/4480*x^7 + 11650311/6400*x^6 - 11531403/1280*x^5 + 6333255/224*x^4 - 120662127/2240*x^3 + 157239459/2800*x^2 - 864291/35*x + 1024
3
11
0:
1
3
11
1:
-3*x + 22
3
11
2:
9/2*x^2 - 129/2*x + 220
3
11
3:
-9/2*x^3 + 189/2*x^2 - 630*x + 1320
3
11
4:
27/8*x^4 - 369/4*x^3 + 7209/8*x^2 - 14769/4*x + 5280
3
11
5:
-81/40*x^5 + 135/2*x^4 - 6867/8*x^3 + 5157*x^2 - 144441/10*x + 14784
3
11
6:
81/80*x^6 - 3159/80*x^5 + 9801/16*x^4 - 76725/16*x^3 + 789237/40*x^2 - 397017/10*x + 29568
3
11
7:
-243/560*x^7 + 1539/80*x^6 - 5589/16*x^5 + 53433/16*x^4 - 717813/40*x^3 + 532737/10*x^2 - 550602/7*x + 42240
3
11
8:
729/4480*x^8 - 8991/1120*x^7 + 10611/64*x^6 - 29727/16*x^5 + 7824303/640*x^4 - 7620957/160*x^3 + 23457519/224*x^2 - 6366951/56*x + 42240
3
11
9:
-243/4480*x^9 + 6561/2240*x^8 - 150903/2240*x^7 + 137619/160*x^6 - 4259061/640*x^5 + 10216287/320*x^4 - 104214051/1120*x^3 + 85928391/560*x^2 - 3388617/28*x + 28160
3
11
10:
729/44800*x^10 - 243/256*x^9 + 15309/640*x^8 - 1526769/4480*x^7 + 19296711/6400*x^6 - 21906207/1280*x^5 + 69532803/1120*x^4 - 44293509/320*x^3 + 69180237/400*x^2 - 3414576/35*x + 11264
3
11
11:
-2187/492800*x^11 + 12393/44800*x^10 - 33777/4480*x^9 + 105705/896*x^8 - 52397361/44800*x^7 + 48921327/6400*x^6 - 74327301/2240*x^5 + 42052635/448*x^4 - 28758312/175*x^3 + 55818981/350*x^2 - 25214547/385*x + 2048
3
12
0:
1
3
12
1:
-3*x + 24
3
12
2:
9/2*x^2 - 141/2*x + 264
3
12
3:
-9/2*x^3 + 207/2*x^2 - 759*x + 1760
3
12
4:
27/8*x^4 - 405/4*x^3 + 8721/8*x^2 - 19809/4*x + 7920
3
12
5:
-81/40*x^5 + 297/4*x^4 - 8343/8*x^3 + 27837/4*x^2 - 109143/5*x + 25344
3
12
6:
81/80*x^6 - 3483/80*x^5 + 11961/16*x^4 - 104193/16*x^3 + 1201797/40*x^2 - 685899/10*x + 59136
3
12
7:
-243/560*x^7 + 1701/80*x^6 - 34263/80*x^5 + 73035/16*x^4 - 550719/20*x^3 + 1854711/20*x^2 - 5532129/35*x + 101376
3
12
8:
729/4480*x^8 - 9963/1120*x^7 + 65367/320*x^6 - 40905/16*x^5 + 12098943/640*x^4 - 13363461/160*x^3 + 236620683/1120*x^2 - 15176583/56*x + 126720
3
12
9:
-243/4480*x^9 + 729/224*x^8 - 186867/2240*x^7 + 95337/80*x^6 - 6637221/640*x^5 + 1804059/32*x^4 - 210907449/1120*x^3 + 101607993/280*x^2 - 2438892/7*x + 112640
3
12
10:
729/44800*x^10 - 9477/8960*x^9 + 133407/4480*x^8 - 2130381/4480*x^7 + 30306231/6400*x^6 - 38942451/1280*x^5 + 35261703/280*x^4 - 726910767/2240*x^3 + 1343545569/2800*x^2 - 23772237/70*x + 67584
3
12
11:
-2187/492800*x^11 + 13851/44800*x^10 - 21141/2240*x^9 + 742851/4480*x^8 - 82932741/44800*x^7 + 87514749/6400*x^6 - 301998051/4480*x^5 + 488394387/2240*x^4 - 2470538799/5600*x^3 + 707537583/1400*x^2 - 100335219/385*x + 24576
3
12
12:
2187/1971200*x^12 - 80919/985600*x^11 + 481869/179200*x^10 - 183951/3584*x^9 + 113344191/179200*x^8 - 471786687/89600*x^7 + 5391736407/179200*x^6 - 2103715557/17920*x^5 + 13642331823/44800*x^4 - 11112782571/22400*x^3 + 27886699377/61600*x^2 - 269917971/1540*x + 4096
3
13
0:
1
3
13
1:
-3*x + 26
3
13
2:
9/2*x^2 - 153/2*x + 312
3
13
3:
-9/2*x^3 + 225/2*x^2 - 900*x + 2288
3
13
4:
27/8*x^4 - 441/4*x^3 + 10377/8*x^2 - 25881/4*x + 11440
3
13
5:
-81/40*x^5 + 81*x^4 - 9963/8*x^3 + 18279/2*x^2 - 317331/10*x + 41184
3
13
6:
81/80*x^6 - 3807/80*x^5 + 14337/16*x^4 - 137565/16*x^3 + 1758537/40*x^2 - 1122471/10*x + 109824
3
13
7:
-243/560*x^7 + 1863/80*x^6 - 41229/80*x^5 + 96957/16*x^4 - 1622403/40*x^3 + 764127/5*x^2 - 10333422/35*x + 219648
3
13
8:
729/4480*x^8 - 2187/224*x^7 + 15795/64*x^6 - 273051/80*x^5 + 17941743/640*x^4 - 4434993/32*x^3 + 88869663/224*x^2 - 164396979/280*x + 329472
3
13
9:
-243/4480*x^9 + 8019/2240*x^8 - 226719/2240*x^7 + 256041/160*x^6 - 9909621/640*x^5 + 30139533/320*x^4 - 397995903/1120*x^3 + 439836669/560*x^2 - 24932151/28*x + 366080
3
13
10:
729/44800*x^10 - 10449/8960*x^9 + 162567/4480*x^8 - 2877849/4480*x^7 + 45560151/6400*x^6 - 13098267/256*x^5 + 133665471/560*x^4 - 1570540563/2240*x^3 + 3375705429/2800*x^2 - 72550077/70*x + 292864
3
13
11:
-2187/492800*x^11 + 2187/6400*x^10 - 51759/4480*x^9 + 201933/896*x^8 - 125540361/44800*x^7 + 148127211/6400*x^6 - 71824401/560*x^5 + 30073761/64*x^4 - 3052546317/2800*x^3 + 256385394/175*x^2 - 361829826/385*x + 159744
3
13
12:
2187/1971200*x^12 - 89667/985600*x^11 + 592677/179200*x^10 - 1258011/17920*x^9 + 24681753/25600*x^8 - 803517651/89600*x^7 + 10292562351/179200*x^6 - 903939993/3584*x^5 + 33178107303/44800*x^4 - 30877092963/22400*x^3 + 12878572383/8800*x^2 - 1072599723/1540*x + 53248
3
13
13:
-6561/25625600*x^13 + 2187/98560*x^12 - 67797/78848*x^11 + 110079/5600*x^10 - 52781301/179200*x^9 + 777843/256*x^8 - 790269777/35840*x^7 + 2529775233/22400*x^6 - 4514231007/11200*x^5 + 2175190497/2240*x^4 - 36633662253/24640*x^3 + 5658359481/4400*x^2 - 4736562333/10010*x + 8192
3
14
0:
1
3
14
1:
-3*x + 28
3
14
2:
9/2*x^2 - 165/2*x + 364
3
14
3:
-9/2*x^3 + 243/2*x^2 - 1053*x + 2912
3
14
4:
27/8*x^4 - 477/4*x^3 + 12177/8*x^2 - 33081/4*x + 16016
3
14
5:
-81/40*x^5 + 351/4*x^4 - 11727/8*x^3 + 46935/4*x^2 - 223368/5*x + 64064
3
14
6:
81/80*x^6 - 4131/80*x^5 + 16929/16*x^4 - 177417/16*x^3 + 2489697/40*x^2 - 1757133/10*x + 192192
3
14
7:
-243/560*x^7 + 405/16*x^6 - 48843/80*x^5 + 125631/16*x^4 - 577557/10*x^3 + 963009/4*x^2 - 18190719/35*x + 439296
3
14
8:
729/4480*x^8 - 1701/160*x^7 + 93879/320*x^6 - 355509/80*x^5 + 25698303/640*x^4 - 35154189/160*x^3 + 786677211/1120*x^2 - 47104533/40*x + 768768
3
14
9:
-243/4480*x^9 + 2187/560*x^8 - 38637/320*x^7 + 41877/20*x^6 - 14278437/640*x^5 + 12020319/80*x^4 - 708445413/1120*x^3 + 110523123/70*x^2 - 20646981/10*x + 1025024
3
14
10:
729/44800*x^10 - 11421/8960*x^9 + 194643/4480*x^8 - 108135/128*x^7 + 66043431/6400*x^6 - 105129819/1280*x^5 + 478307673/1120*x^4 - 632504835/448*x^3 + 7774072119/2800*x^2 - 14086488/5*x + 1025024
3
14
11:
-2187/492800*x^11 + 16767/44800*x^10 - 486/35*x^9 + 1334799/4480*x^8 - 26156763/6400*x^7 + 239247513/6400*x^6 - 1033034553/4480*x^5 + 2121905403/2240*x^4 - 13957795449/5600*x^3 + 5426788581/1400*x^2 - 165697239/55*x + 745472
3
14
12:
2187/1971200*x^12 - 19683/197120*x^11 + 715149/179200*x^10 - 238869/2560*x^9 + 253545471/179200*x^8 - 37305117/2560*x^7 + 18587686167/179200*x^6 - 9116461629/17920*x^5 + 75281372703/44800*x^4 - 2277652401/640*x^3 + 270645324057/61600*x^2 - 566748333/220*x + 372736
3
14
13:
-6561/25625600*x^13 + 2187/89600*x^12 - 2053593/1971200*x^11 + 2353941/89600*x^10 - 11134503/25600*x^9 + 63573903/12800*x^8 - 7165419489/179200*x^7 + 20411663283/89600*x^6 - 40655423853/44800*x^5 + 54930012273/22400*x^4 - 37344023847/8800*x^3 + 210646953/50*x^2 - 1334311338/715*x + 114688
3
14
14:
19683/358758400*x^14 - 282123/51251200*x^13 + 981963/3942400*x^12 - 26482383/3942400*x^11 + 43083171/358400*x^10 - 538906203/358400*x^9 + 33832596213/2508800*x^8 - 31357977039/358400*x^7 + 73276017147/179200*x^6 - 121178116449/89600*x^5 + 1506308546961/492800*x^4 - 273055533417/61600*x^3 + 10253888308359/2802800*x^2 - 1841849643/1430*x + 16384
3
15
0:
1
3
15
1:
-3*x + 30
3
15
2:
9/2*x^2 - 177/2*x + 420
3
15
3:
-9/2*x^3 + 261/2*x^2 - 1218*x + 3640
3
15
4:
27/8*x^4 - 513/4*x^3 + 14121/8*x^2 - 41505/4*x + 21840
3
15
5:
-81/40*x^5 + 189/2*x^4 - 13635/8*x^3 + 14778*x^2 - 612141/10*x + 96096
3
15
6:
81/80*x^6 - 891/16*x^5 + 19737/16*x^4 - 224325/16*x^3 + 3428397/40*x^2 - 530121/2*x + 320320
3
15
7:
-243/560*x^7 + 2187/80*x^6 - 11421/16*x^5 + 159489/16*x^4 - 3197313/40*x^3 + 3652371/10*x^2 - 6098130/7*x + 823680
3
15
8:
729/4480*x^8 - 12879/1120*x^7 + 110079/320*x^6 - 90639/16*x^5 + 35748783/640*x^4 - 53636013/160*x^3 + 1325962251/1120*x^2 - 124156647/56*x + 1647360
3
15
9:
-243/4480*x^9 + 9477/2240*x^8 - 45441/320*x^7 + 85779/32*x^6 - 19966581/640*x^5 + 73779579/320*x^4 - 1200604059/1120*x^3 + 334172439/112*x^2 - 17679699/4*x + 2562560
3
15
10:
729/44800*x^10 - 12393/8960*x^9 + 6561/128*x^8 - 695223/640*x^7 + 92844711/6400*x^6 - 162243567/1280*x^5 + 162975321/224*x^4 - 5996305827/2240*x^3 + 2373703137/400*x^2 - 34733469/5*x + 3075072
3
15
11:
-2187/492800*x^11 + 729/1792*x^10 - 73629/4480*x^9 + 344817/896*x^8 - 36970263/6400*x^7 + 14853375/256*x^6 - 884471643/2240*x^5 + 807027219/448*x^4 - 7442604081/1400*x^3 + 132008607/14*x^2 - 95119995/11*x + 2795520
3
15
12:
2187/1971200*x^12 - 15309/140800*x^11 + 169857/35840*x^10 - 2169747/17920*x^9 + 360329391/179200*x^8 - 291152637/12800*x^7 + 6397109379/35840*x^6 - 2482962579/2560*x^5 + 160157588823/44800*x^4 - 191380197627/22400*x^3 + 149640543837/12320*x^2 - 378465249/44*x + 1863680
3
15
13:
-6561/25625600*x^13 + 6561/246400*x^12 - 2447253/1971200*x^11 + 306909/8960*x^10 - 15911883/25600*x^9 + 87320349/11200*x^8 - 12388135869/179200*x^7 + 779986989/1792*x^6 - 43118865999/22400*x^5 + 8138211561/1400*x^4 - 199958929749/17600*x^3 + 80080769421/6160*x^2 - 2007270201/286*x + 860160
3
15
14:
19683/358758400*x^14 - 308367/51251200*x^13 + 1174419/3942400*x^12 - 6939351/788480*x^11 + 8844957/51200*x^10 - 850672287/358400*x^9 + 58753566189/2508800*x^8 - 12003930999/71680*x^7 + 154922670279/179200*x^6 - 283799811861/89600*x^5 + 560461298139/70400*x^4 - 31834874691/2464*x^3 + 6773987900619/560560*x^2 - 1435818999/286*x + 245760
3
15
15:
-19683/1793792000*x^15 + 452709/358758400*x^14 - 675783/10250240*x^13 + 8164071/3942400*x^12 - 122797863/2816000*x^11 + 232903107/358400*x^10 - 3537313821/501760*x^9 + 141482052081/2508800*x^8 - 298457569107/896000*x^7 + 128985560001/89600*x^6 - 62761162851/14080*x^5 + 586783236069/61600*x^4 - 46135891636281/3503500*x^3 + 1461283700913/140140*x^2 - 505162842/143*x + 32768
3
16
0:
1
3
16
1:
-3*x + 32
3
16
2:
9/2*x^2 - 189/2*x + 480
3
16
3:
-9/2*x^3 + 279/2*x^2 - 1395*x + 4480
3
16
4:
27/8*x^4 - 549/4*x^3 + 16209/8*x^2 - 51249/4*x + 29120
3
16
5:
-81/40*x^5 + 405/4*x^4 - 15687/8*x^3 + 73233/4*x^2 - 409833/5*x + 139776
3
16
6:
81/80*x^6 - 4779/80*x^5 + 22761/16*x^4 - 278865/16*x^3 + 4610637/40*x^2 - 3874887/10*x + 512512
3
16
7:
-243/560*x^7 + 2349/80*x^6 - 13203/16*x^5 + 198963/16*x^4 - 2159469/20*x^3 + 10733139/20*x^2 - 9808977/7*x + 1464320
3
16
8:
729/4480*x^8 - 13851/1120*x^7 + 25515/64*x^6 - 113481/16*x^5 + 48507903/640*x^4 - 79214517/160*x^3 + 428818671/224*x^2 - 221726727/56*x + 3294720
3
16
9:
-243/4480*x^9 + 729/160*x^8 - 369603/2240*x^7 + 269487/80*x^6 - 27217701/640*x^5 + 54764181/160*x^4 - 1951508241/1120*x^3 + 214058889/40*x^2 - 61978635/7*x + 5857280
3
16
10:
729/44800*x^10 - 2673/1792*x^9 + 267543/4480*x^8 - 876987/640*x^7 + 127156311/6400*x^6 - 242109891/1280*x^5 + 665666829/560*x^4 - 10798722063/2240*x^3 + 33324543909/2800*x^2 - 157865433/10*x + 8200192
3
16
11:
-2187/492800*x^11 + 19683/44800*x^10 - 43011/2240*x^9 + 436671/896*x^8 - 50874723/6400*x^7 + 557023797/6400*x^6 - 580929651/896*x^5 + 1458928503/448*x^4 - 59751945459/5600*x^3 + 29816782659/1400*x^2 - 1239736293/55*x + 8945664
3
16
12:
2187/1971200*x^12 - 115911/985600*x^11 + 28431/5120*x^10 - 2758779/17920*x^9 + 498256191/179200*x^8 - 439033689/12800*x^7 + 10556054379/35840*x^6 - 31532284341/17920*x^5 + 45937576089/6400*x^4 - 429543528219/22400*x^3 + 381975692157/12320*x^2 - 1139425209/44*x + 7454720
3
16
13:
-6561/25625600*x^13 + 28431/985600*x^12 - 575181/394240*x^11 + 156735/3584*x^10 - 154778121/179200*x^9 + 1058892183/89600*x^8 - 4108081941/35840*x^7 + 14196979269/17920*x^6 - 173141422263/44800*x^5 + 290368973799/22400*x^4 - 175244734107/6160*x^3 + 114860656629/3080*x^2 - 3463659219/143*x + 4587520
3
16
14:
19683/358758400*x^14 - 334611/51251200*x^13 + 1384371/3942400*x^12 - 44485767/3942400*x^11 + 86467419/358400*x^10 - 1296205011/358400*x^9 + 97873082541/2508800*x^8 - 109572198471/358400*x^7 + 310920068079/179200*x^6 - 628750739853/89600*x^5 + 9652530025917/492800*x^4 - 1097792187759/30800*x^3 + 21348687935241/560560*x^2 - 5450359401/286*x + 1966080
3
16
15:
-19683/1793792000*x^15 + 19683/14350336*x^14 - 570807/7321600*x^13 + 955719/358400*x^12 - 1206552591/19712000*x^11 + 71346501/71680*x^10 - 29595981123/2508800*x^9 + 258989184459/2508800*x^8 - 42753988863/64000*x^7 + 7097705757/2240*x^6 - 332402414391/30800*x^5 + 6270362031249/246400*x^4 - 546665266155249/14014000*x^3 + 176301005499/5096*x^2 - 1940981841/143*x + 524288
3
16
16:
59049/28700672000*x^16 - 19683/73216000*x^15 + 23048793/1435033600*x^14 - 1489347/2562560*x^13 + 320397687/22528000*x^12 - 4905158877/19712000*x^11 + 32204505933/10035200*x^10 - 555270309/17920*x^9 + 45195964479639/200704000*x^8 - 4399243568271/3584000*x^7 + 5582944953297/1126400*x^6 - 2855077815627/197120*x^5 + 52837414375852521/1793792000*x^4 - 1249973005469229/32032000*x^3 + 133557037496163/4484480*x^2 - 22320892047/2288*x + 65536
4
1
0:
1
4
1
1:
-4*x + 3
4
2
0:
1
4
2
1:
-4*x + 6
4
2
2:
8*x^2 - 20*x + 9
4
3
0:
1
4
3
1:
-4*x + 9
4
3
2:
8*x^2 - 32*x + 27
4
3
3:
-32/3*x^3 + 56*x^2 - 244/3*x + 27
4
4
0:
1
4
4
1:
-4*x + 12
4
4
2:
8*x^2 - 44*x + 54
4
4
3:
-32/3*x^3 + 80*x^2 - 532/3*x + 108
4
4
4:
32/3*x^4 - 96*x^3 + 856/3*x^2 - 308*x + 81
4
5
0:
1
4
5
1:
-4*x + 15
4
5
2:
8*x^2 - 56*x + 90
4
5
3:
-32/3*x^3 + 104*x^2 - 928/3*x + 270
4
5
4:
32/3*x^4 - 128*x^3 + 1576/3*x^2 - 840*x + 405
4
5
5:
-128/15*x^5 + 352/3*x^4 - 1760/3*x^3 + 3848/3*x^2 - 5644/5*x + 243
4
6
0:
1
4
6
1:
-4*x + 18
4
6
2:
8*x^2 - 68*x + 135
4
6
3:
-32/3*x^3 + 128*x^2 - 1432/3*x + 540
4
6
4:
32/3*x^4 - 160*x^3 + 2512/3*x^2 - 1768*x + 1215
4
6
5:
-128/15*x^5 + 448/3*x^4 - 2912/3*x^3 + 8576/3*x^2 - 18244/5*x + 1458
4
6
6:
256/45*x^6 - 1664/15*x^5 + 7520/9*x^4 - 3040*x^3 + 243304/45*x^2 - 61036/15*x + 729
4
7
0:
1
4
7
1:
-4*x + 21
4
7
2:
8*x^2 - 80*x + 189
4
7
3:
-32/3*x^3 + 152*x^2 - 2044/3*x + 945
4
7
4:
32/3*x^4 - 192*x^3 + 3664/3*x^2 - 3200*x + 2835
4
7
5:
-128/15*x^5 + 544/3*x^4 - 4352/3*x^3 + 16112/3*x^2 - 44764/5*x + 5103
4
7
6:
256/45*x^6 - 2048/15*x^5 + 11552/9*x^4 - 5952*x^3 + 629224/45*x^2 - 225232/15*x + 5103
4
7
7:
-1024/315*x^7 + 256/3*x^6 - 40576/45*x^5 + 4896*x^4 - 647968/45*x^3 + 65864/3*x^2 - 509172/35*x + 2187
4
8
0:
1
4
8
1:
-4*x + 24
4
8
2:
8*x^2 - 92*x + 252
4
8
3:
-32/3*x^3 + 176*x^2 - 2764/3*x + 1512
4
8
4:
32/3*x^4 - 224*x^3 + 5032/3*x^2 - 5244*x + 5670
4
8
5:
-128/15*x^5 + 640/3*x^4 - 6080/3*x^3 + 27104/3*x^2 - 92764/5*x + 13608
4
8
6:
256/45*x^6 - 2432/15*x^5 + 16448/9*x^4 - 10304*x^3 + 1354264/45*x^2 - 628108/15*x + 20412
4
8
7:
-1024/315*x^7 + 512/5*x^6 - 59008/45*x^5 + 26240/3*x^4 - 1451488/45*x^3 + 958544/15*x^2 - 2085796/35*x + 17496
4
8
8:
512/315*x^8 - 17408/315*x^7 + 35072/45*x^6 - 265088/45*x^5 + 1156064/45*x^4 - 2904416/45*x^3 + 3048664/35*x^2 - 1814236/35*x + 6561
4
9
0:
1
4
9
1:
-4*x + 27
4
9
2:
8*x^2 - 104*x + 324
4
9
3:
-32/3*x^3 + 200*x^2 - 3592/3*x + 2268
4
9
4:
32/3*x^4 - 256*x^3 + 6616/3*x^2 - 8008*x + 10206
4
9
5:
-128/15*x^5 + 736/3*x^4 - 8096/3*x^3 + 42200/3*x^2 - 171424/5*x + 30618
4
9
6:
256/45*x^6 - 2816/15*x^5 + 22208/9*x^4 - 16384*x^3 + 2573944/45*x^2 - 1462984/15*x + 61236
4
9
7:
-1024/315*x^7 + 1792/15*x^6 - 80896/45*x^5 + 42688/3*x^4 - 2842528/45*x^3 + 770936/5*x^2 - 6482552/35*x + 78732
4
9
8:
512/315*x^8 - 4096/63*x^7 + 48896/45*x^6 - 442112/45*x^5 + 2336864/45*x^4 - 1451776/9*x^3 + 9758472/35*x^2 - 8071624/35*x + 59049
4
9
9:
-2048/2835*x^9 + 9728/315*x^8 - 529408/945*x^7 + 84224/15*x^6 - 4575104/135*x^5 + 628384/5*x^4 - 790852576/2835*x^3 + 107250376/315*x^2 - 58159412/315*x + 19683
4
10
0:
1
4
10
1:
-4*x + 30
4
10
2:
8*x^2 - 116*x + 405
4
10
3:
-32/3*x^3 + 224*x^2 - 4528/3*x + 3240
4
10
4:
32/3*x^4 - 288*x^3 + 8416/3*x^2 - 11600*x + 17010
4
10
5:
-128/15*x^5 + 832/3*x^4 - 10400/3*x^3 + 62048/3*x^2 - 291544/5*x + 61236
4
10
6:
256/45*x^6 - 640/3*x^5 + 28832/9*x^4 - 24480*x^3 + 4472944/45*x^2 - 601160/3*x + 153090
4
10
7:
-1024/315*x^7 + 2048/15*x^6 - 21248/9*x^5 + 21632*x^4 - 5054368/45*x^3 + 4886752/15*x^2 - 3344688/7*x + 262440
4
10
8:
512/315*x^8 - 23552/315*x^7 + 65024/45*x^6 - 136960/9*x^5 + 4257824/45*x^4 - 15786464/45*x^3 + 25948128/35*x^2 - 5503856/7*x + 295245
4
10
9:
-2048/2835*x^9 + 11264/315*x^8 - 713728/945*x^7 + 26624/3*x^6 - 8554112/135*x^5 + 4222016/15*x^4 - 2162780896/2835*x^3 + 74145824/63*x^2 - 55218652/63*x + 196830
4
10
10:
4096/14175*x^10 - 2048/135*x^9 + 65024/189*x^8 - 1389568/315*x^7 + 23711488/675*x^6 - 8077184/45*x^5 + 332343968/567*x^4 - 158725408/135*x^3 + 2075961784/1575*x^2 - 13833892/21*x + 59049
4
11
0:
1
4
11
1:
-4*x + 33
4
11
2:
8*x^2 - 128*x + 495
4
11
3:
-32/3*x^3 + 248*x^2 - 5572/3*x + 4455
4
11
4:
32/3*x^4 - 320*x^3 + 10432/3*x^2 - 16128*x + 26730
4
11
5:
-128/15*x^5 + 928/3*x^4 - 12992/3*x^3 + 87296/3*x^2 - 465544/5*x + 112266
4
11
6:
256/45*x^6 - 3584/15*x^5 + 36320/9*x^4 - 34880*x^3 + 7265104/45*x^2 - 5629696/15*x + 336798
4
11
7:
-1024/315*x^7 + 768/5*x^6 - 27008/9*x^5 + 93728/3*x^4 - 8359168/45*x^3 + 9359696/15*x^2 - 7552808/7*x + 721710
4
11
8:
512/315*x^8 - 26624/315*x^7 + 83456/45*x^6 - 200704/9*x^5 + 7178144/45*x^4 - 30949568/45*x^3 + 60155392/35*x^2 - 15537920/7*x + 1082565
4
11
9:
-2048/2835*x^9 + 2560/63*x^8 - 925696/945*x^7 + 66048/5*x^6 - 14717312/135*x^5 + 1695968/3*x^4 - 5146422592/2835*x^3 + 1071328576/315*x^2 - 203822764/63*x + 1082565
4
11
10:
4096/14175*x^10 - 16384/945*x^9 + 60928/135*x^8 - 2103296/315*x^7 + 41682688/675*x^6 - 16631296/45*x^5 + 4055602912/2835*x^4 - 3273858752/945*x^3 + 1090985512/225*x^2 - 69052544/21*x + 649539
4
11
11:
-16384/155925*x^11 + 94208/14175*x^10 - 34816/189*x^9 + 79360/27*x^8 - 140862464/4725*x^7 + 135113984/675*x^6 - 508892032/567*x^5 + 1498054624/567*x^4 - 2555347232/525*x^3 + 1140981656/225*x^2 - 181532940/77*x + 177147
4
12
0:
1
4
12
1:
-4*x + 36
4
12
2:
8*x^2 - 140*x + 594
4
12
3:
-32/3*x^3 + 272*x^2 - 6724/3*x + 5940
4
12
4:
32/3*x^4 - 352*x^3 + 12664/3*x^2 - 21700*x + 40095
4
12
5:
-128/15*x^5 + 1024/3*x^4 - 15872/3*x^3 + 118592/3*x^2 - 707464/5*x + 192456
4
12
6:
256/45*x^6 - 3968/15*x^5 + 44672/9*x^4 - 47872*x^3 + 11193424/45*x^2 - 9819592/15*x + 673596
4
12
7:
-1024/315*x^7 + 512/3*x^6 - 167296/45*x^5 + 130048/3*x^4 - 13067968/45*x^3 + 1108320*x^2 - 77171912/35*x + 1732104
4
12
8:
512/315*x^8 - 29696/315*x^7 + 104192/45*x^6 - 281728/9*x^5 + 11395904/45*x^4 - 56027072/45*x^3 + 125673264/35*x^2 - 38196344/7*x + 3247695
4
12
9:
-2048/2835*x^9 + 2048/45*x^8 - 1165312/945*x^7 + 56320/3*x^6 - 23748992/135*x^5 + 5219328/5*x^4 - 10995890944/2835*x^3 + 77014976/9*x^2 - 623346604/63*x + 4330260
4
12
10:
4096/14175*x^10 - 2048/105*x^9 + 541696/945*x^8 - 1009664/105*x^7 + 68432128/675*x^6 - 10449536/15*x^5 + 8863672192/2835*x^4 - 311862272/35*x^3 + 23706827224/1575*x^2 - 90958436/7*x + 3897234
4
12
11:
-16384/155925*x^11 + 106496/14175*x^10 - 223232/945*x^9 + 579584/135*x^8 - 235510784/4725*x^7 + 260162048/675*x^6 - 5687775104/2835*x^5 + 19657081856/2835*x^4 - 24035335456/1575*x^3 + 4413938192/225*x^2 - 941110924/77*x + 2125764
4
12
12:
16384/467775*x^12 - 16384/6237*x^11 + 3715072/42525*x^10 - 59392/35*x^9 + 302447104/14175*x^8 - 171817984/945*x^7 + 45405174016/42525*x^6 - 12200617088/2835*x^5 + 493839410144/42525*x^4 - 18794107616/945*x^3 + 1010143651448/51975*x^2 - 1956757868/231*x + 531441
4
13
0:
1
4
13
1:
-4*x + 39
4
13
2:
8*x^2 - 152*x + 702
4
13
3:
-32/3*x^3 + 296*x^2 - 7984/3*x + 7722
4
13
4:
32/3*x^4 - 384*x^3 + 15112/3*x^2 - 28424*x + 57915
4
13
5:
-128/15*x^5 + 1120/3*x^4 - 19040/3*x^3 + 156584/3*x^2 - 1032964/5*x + 312741
4
13
6:
256/45*x^6 - 4352/15*x^5 + 53888/9*x^4 - 63744*x^3 + 16530064/45*x^2 - 16186768/15*x + 1250964
4
13
7:
-1024/315*x^7 + 2816/15*x^6 - 203008/45*x^5 + 58240*x^4 - 19530688/45*x^3 + 27818224/15*x^2 - 145909056/35*x + 3752892
4
13
8:
512/315*x^8 - 32768/315*x^7 + 127232/45*x^6 - 1910528/45*x^5 + 17248064/45*x^4 - 95230976/45*x^3 + 242046864/35*x^2 - 422497456/35*x + 8444007
4
13
9:
-2048/2835*x^9 + 15872/315*x^8 - 1432576/945*x^7 + 385792/15*x^6 - 36426752/135*x^5 + 27053888/15*x^4 - 21585007552/2835*x^3 + 6088702288/315*x^2 - 1654647892/63*x + 14073345
4
13
10:
4096/14175*x^10 - 4096/189*x^9 + 134144/189*x^8 - 4194304/315*x^7 + 106448128/675*x^6 - 11019520/9*x^5 + 3548349824/567*x^4 - 19416172288/945*x^3 + 64139689624/1575*x^2 - 896221912/21*x + 16888014
4
13
11:
-16384/155925*x^11 + 118784/14175*x^10 - 278528/945*x^9 + 5682176/945*x^8 - 371815424/4725*x^7 + 465458432/675*x^6 - 11612662016/2835*x^5 + 46248098432/2835*x^4 - 66136742176/1575*x^3 + 102018049016/1575*x^2 - 3942739312/77*x + 13817466
4
13
12:
16384/467775*x^12 - 65536/22275*x^11 + 667648/6075*x^10 - 151552/63*x^9 + 485016064/14175*x^8 - 1565622272/4725*x^7 + 94575801088/42525*x^6 - 836112640/81*x^5 + 196915441952/6075*x^4 - 310288557184/4725*x^3 + 4069002818504/51975*x^2 - 10426756184/231*x + 6908733
4
13
13:
-65536/6081075*x^13 + 16384/17325*x^12 - 17383424/467775*x^11 + 12234752/14175*x^10 - 186652672/14175*x^9 + 655161856/4725*x^8 - 43814167552/42525*x^7 + 25628463872/4725*x^6 - 850221525632/42525*x^5 + 709850784736/14175*x^4 - 4182965250656/51975*x^3 + 1287680747128/17325*x^2 - 30605234812/1001*x + 1594323
4
14
0:
1
4
14
1:
-4*x + 42
4
14
2:
8*x^2 - 164*x + 819
4
14
3:
-32/3*x^3 + 320*x^2 - 9352/3*x + 9828
4
14
4:
32/3*x^4 - 416*x^3 + 17776/3*x^2 - 36408*x + 81081
4
14
5:
-128/15*x^5 + 1216/3*x^4 - 22496/3*x^3 + 201920/3*x^2 - 1459324/5*x + 486486
4
14
6:
256/45*x^6 - 4736/15*x^5 + 63968/9*x^4 - 82784*x^3 + 23576344/45*x^2 - 25483444/15*x + 2189187
4
14
7:
-1024/315*x^7 + 1024/5*x^6 - 242176/45*x^5 + 228608/3*x^4 - 28136128/45*x^3 + 44348288/15*x^2 - 259216432/35*x + 7505784
4
14
8:
512/315*x^8 - 1024/9*x^7 + 152576/45*x^6 - 2519552/45*x^5 + 25110464/45*x^4 - 30764608/9*x^3 + 436774432/35*x^2 - 122889232/5*x + 19702683
4
14
9:
-2048/2835*x^9 + 17408/315*x^8 - 246784/135*x^7 + 171008/5*x^6 - 53621504/135*x^5 + 44301952/15*x^4 - 39583662016/2835*x^3 + 12623967616/315*x^2 - 2811524396/45*x + 39405366
4
14
10:
4096/14175*x^10 - 22528/945*x^9 + 116224/135*x^8 - 160768/9*x^7 + 158530048/675*x^6 - 91524352/45*x^5 + 33081303616/2835*x^4 - 8200235968/189*x^3 + 22210031992/225*x^2 - 364409972/3*x + 59108049
4
14
11:
-16384/155925*x^11 + 131072/14175*x^10 - 339968/945*x^9 + 7694336/945*x^8 - 80079872/675*x^7 + 784802816/675*x^6 - 22026108416/2835*x^5 + 99473345792/2835*x^4 - 54405867872/525*x^3 + 294437117888/1575*x^2 - 1971597192/11*x + 64481508
4
14
12:
16384/467775*x^12 - 507904/155925*x^11 + 5742592/42525*x^10 - 1036288/315*x^9 + 740713984/14175*x^8 - 383009792/675*x^7 + 182547444736/42525*x^6 - 64101928448/2835*x^5 + 3459572523104/42525*x^4 - 905519236768/4725*x^3 + 14168789671088/51975*x^2 - 6558772856/33*x + 48361131
4
14
13:
-65536/6081075*x^13 + 32768/31185*x^12 - 21512192/467775*x^11 + 5636096/4725*x^10 - 288950272/14175*x^9 + 32576512/135*x^8 - 86085968896/42525*x^7 + 171461192704/14175*x^6 - 2167098933632/42525*x^5 + 27843451712/189*x^4 - 14422487637728/51975*x^3 + 85026723264/275*x^2 - 23736152172/143*x + 22320522
4
14
14:
131072/42567525*x^14 - 1900544/6081075*x^13 + 6701056/467775*x^12 - 183353344/467775*x^11 + 101060608/14175*x^10 - 1287538688/14175*x^9 + 247598322176/297675*x^8 - 235009436672/42525*x^7 + 1128803451136/42525*x^6 - 3854536851328/42525*x^5 + 33178058481632/155925*x^4 - 16805409668896/51975*x^3 + 447669704443384/1576575*x^2 - 111008839540/1001*x + 4782969
4
15
0:
1
4
15
1:
-4*x + 45
4
15
2:
8*x^2 - 176*x + 945
4
15
3:
-32/3*x^3 + 344*x^2 - 10828/3*x + 12285
4
15
4:
32/3*x^4 - 448*x^3 + 20656/3*x^2 - 45760*x + 110565
4
15
5:
-128/15*x^5 + 1312/3*x^4 - 26240/3*x^3 + 255248/3*x^2 - 2005444/5*x + 729729
4
15
6:
256/45*x^6 - 1024/3*x^5 + 74912/9*x^4 - 105280*x^3 + 32662744/45*x^2 - 7723472/3*x + 3648645
4
15
7:
-1024/315*x^7 + 3328/15*x^6 - 56960/9*x^5 + 292576/3*x^4 - 39311968/45*x^3 + 22641544/5*x^2 - 87520108/7*x + 14073345
4
15
8:
512/315*x^8 - 38912/315*x^7 + 180224/45*x^6 - 649216/9*x^5 + 35397824/45*x^4 - 238231424/45*x^3 + 747212448/35*x^2 - 327574784/7*x + 42220035
4
15
9:
-2048/2835*x^9 + 18944/315*x^8 - 292864/135*x^7 + 133120/3*x^6 - 76297472/135*x^5 + 23137472/5*x^4 - 68656216576/2835*x^3 + 4883375456/63*x^2 - 1225906732/9*x + 98513415
4
15
10:
4096/14175*x^10 - 8192/315*x^9 + 194048/189*x^8 - 116736/5*x^7 + 227788288/675*x^6 - 48381952/15*x^5 + 11640102080/567*x^4 - 26861613952/315*x^3 + 344829738184/1575*x^2 - 1544524752/5*x + 177324147
4
15
11:
-16384/155925*x^11 + 4096/405*x^10 - 407552/945*x^9 + 2027008/189*x^8 - 116252672/675*x^7 + 252078592/135*x^6 - 39324210944/2835*x^5 + 5677635904/81*x^4 - 368223502816/1575*x^3 + 152169557944/315*x^2 - 5980106884/11*x + 241805655
4
15
12:
16384/467775*x^12 - 557056/155925*x^11 + 1384448/8505*x^10 - 65536/15*x^9 + 155279872/2025*x^8 - 623249408/675*x^7 + 66175035392/8505*x^6 - 130180253696/2835*x^5 + 7935873083744/42525*x^4 - 339211095616/675*x^3 + 247531796240/297*x^2 - 24303147584/33*x + 241805655
4
15
13:
-65536/6081075*x^13 + 16384/14175*x^12 - 26083328/467775*x^11 + 647168/405*x^10 - 428849152/14175*x^9 + 626963968/1575*x^8 - 158474819584/42525*x^7 + 70801727488/2835*x^6 - 5051685713792/42525*x^5 + 792547343072/2025*x^4 - 44304622451072/51975*x^3 + 355008604304/315*x^2 - 109000199300/143*x + 167403915
4
15
14:
131072/42567525*x^14 - 2097152/6081075*x^13 + 8175616/467775*x^12 - 49577984/93555*x^11 + 151785472/14175*x^10 - 2154389504/14175*x^9 + 463091949056/297675*x^8 - 98653468672/8505*x^7 + 2671954185472/42525*x^6 - 10355833652224/42525*x^5 + 102090601468832/155925*x^4 - 12014574516416/10395*x^3 + 382008863572184/315315*x^2 - 609468035152/1001*x + 71744535
4
15
15:
-524288/638512875*x^15 + 4063232/42567525*x^14 - 18415616/3648645*x^13 + 2277376/14175*x^12 - 3441901568/1002375*x^11 + 2212401152/42525*x^10 - 20539598848/35721*x^9 + 1398204773888/297675*x^8 - 18122512661504/637875*x^7 + 23844266752/189*x^6 - 16161239162752/40095*x^5 + 418051564691168/467775*x^4 - 91662941840523488/70945875*x^3 + 466354429778984/429975*x^2 - 6070213345124/15015*x + 14348907
4
16
0:
1
4
16
1:
-4*x + 48
4
16
2:
8*x^2 - 188*x + 1080
4
16
3:
-32/3*x^3 + 368*x^2 - 12412/3*x + 15120
4
16
4:
32/3*x^4 - 480*x^3 + 23752/3*x^2 - 56588*x + 147420
4
16
5:
-128/15*x^5 + 1408/3*x^4 - 30272/3*x^3 + 317216/3*x^2 - 2691844/5*x + 1061424
4
16
6:
256/45*x^6 - 5504/15*x^5 + 86720/9*x^4 - 131520*x^3 + 44148904/45*x^2 - 56666356/15*x + 5837832
4
16
7:
-1024/315*x^7 + 3584/15*x^6 - 66176/9*x^5 + 122496*x^4 - 53524768/45*x^3 + 100587376/15*x^2 - 141584412/7*x + 25019280
4
16
8:
512/315*x^8 - 41984/315*x^7 + 210176/45*x^6 - 820096/9*x^5 + 48563744/45*x^4 - 356167328/45*x^3 + 1222684872/35*x^2 - 590135108/7*x + 84440070
4
16
9:
-2048/2835*x^9 + 4096/63*x^8 - 2400256/945*x^7 + 845824/15*x^6 - 105512192/135*x^5 + 20962048/3*x^4 - 113681955712/2835*x^3 + 44591613376/315*x^2 - 17425866292/63*x + 225173520
4
16
10:
4096/14175*x^10 - 26624/945*x^9 + 1140736/945*x^8 - 1343488/45*x^7 + 317644288/675*x^6 - 221443328/45*x^5 + 97557350272/2835*x^4 - 149241058432/945*x^3 + 711082897384/1575*x^2 - 10763107916/15*x + 472864392
4
16
11:
-16384/155925*x^11 + 155648/14175*x^10 - 96256/189*x^9 + 372736/27*x^8 - 163530752/675*x^7 + 1943757824/675*x^6 - 66756777728/2835*x^5 + 74663757568/567*x^4 - 771147712096/1575*x^3 + 256476714896/225*x^2 - 80869851236/55*x + 773778096
4
16
12:
16384/467775*x^12 - 606208/155925*x^11 + 1642496/8505*x^10 - 1783808/315*x^9 + 1543035904/14175*x^8 - 972007424/675*x^7 + 22763577856/1701*x^6 - 248152886528/2835*x^5 + 16878149632544/42525*x^4 - 5688489194656/4725*x^3 + 23728399104856/10395*x^2 - 78124109540/33*x + 967222620
4
16
13:
-65536/6081075*x^13 + 65536/51975*x^12 - 31096832/467775*x^11 + 5914624/2835*x^10 - 614643712/14175*x^9 + 2967851008/4725*x^8 - 276268957696/42525*x^7 + 1014642688/21*x^6 - 10909797130112/42525*x^5 + 13483704485248/14175*x^4 - 122662385538368/51975*x^3 + 12568707515744/3465*x^2 - 424941117892/143*x + 892820880
4
16
14:
131072/42567525*x^14 - 65536/173745*x^13 + 9797632/467775*x^12 - 326139904/467775*x^11 + 219738112/14175*x^10 - 688187392/2835*x^9 + 818580518912/297675*x^8 - 968691802112/42525*x^7 + 5858031922432/42525*x^6 - 1020435631744/1701*x^5 + 285169037718464/155925*x^4 - 192986739935296/51975*x^3 + 1448099702297096/315315*x^2 - 2898472220452/1001*x + 573956280
4
16
15:
-524288/638512875*x^15 + 4456448/42567525*x^14 - 110952448/18243225*x^13 + 11075584/51975*x^12 - 35248357376/7016625*x^11 + 143138816/1701*x^10 - 920669587456/893025*x^9 + 2787480621056/297675*x^8 - 40319543112704/637875*x^7 + 4460274191872/14175*x^6 - 1590870902266496/1403325*x^5 + 1336866977910656/467775*x^4 - 337661355064141088/70945875*x^3 + 22320297588317104/4729725*x^2 - 33496274926964/15015*x + 229582512
4
16
16:
131072/638512875*x^16 - 524288/19348875*x^15 + 3801088/2321865*x^14 - 3473408/57915*x^13 + 10439081984/7016625*x^12 - 61810589696/2338875*x^11 + 309517447168/893025*x^10 - 40600954880/11907*x^9 + 112993420508672/4465125*x^8 - 10009853252608/70875*x^7 + 821987618646272/1403325*x^6 - 3002039666560/1701*x^5 + 23987390891938912/6449625*x^4 - 121595017632989408/23648625*x^3 + 2178067290214888/525525*x^2 - 7413732802404/5005*x + 43046721
5
1
0:
1
5
1
1:
-5*x + 4
5
2
0:
1
5
2
1:
-5*x + 8
5
2
2:
25/2*x^2 - 65/2*x + 16
5
3
0:
1
5
3
1:
-5*x + 12
5
3
2:
25/2*x^2 - 105/2*x + 48
5
3
3:
-125/6*x^3 + 225/2*x^2 - 515/3*x + 64
5
4
0:
1
5
4
1:
-5*x + 16
5
4
2:
25/2*x^2 - 145/2*x + 96
5
4
3:
-125/6*x^3 + 325/2*x^2 - 1145/3*x + 256
5
4
4:
625/24*x^4 - 2875/12*x^3 + 17675/24*x^2 - 10115/12*x + 256
5
5
0:
1
5
5
1:
-5*x + 20
5
5
2:
25/2*x^2 - 185/2*x + 160
5
5
3:
-125/6*x^3 + 425/2*x^2 - 2015/3*x + 640
5
5
4:
625/24*x^4 - 3875/12*x^3 + 33275/24*x^2 - 28435/12*x + 1280
5
5
5:
-625/24*x^5 + 4375/12*x^4 - 44875/24*x^3 + 50975/12*x^2 - 11990/3*x + 1024
5
6
0:
1
5
6
1:
-5*x + 24
5
6
2:
25/2*x^2 - 225/2*x + 240
5
6
3:
-125/6*x^3 + 525/2*x^2 - 3125/3*x + 1280
5
6
4:
625/24*x^4 - 1625/4*x^3 + 53675/24*x^2 - 20225/4*x + 3840
5
6
5:
-625/24*x^5 + 1875/4*x^4 - 75875/24*x^3 + 39175/4*x^2 - 13475*x + 6144
5
6
6:
3125/144*x^6 - 6875/16*x^5 + 475625/144*x^4 - 593375/48*x^3 + 1651525/72*x^2 - 111545/6*x + 4096
5
7
0:
1
5
7
1:
-5*x + 28
5
7
2:
25/2*x^2 - 265/2*x + 336
5
7
3:
-125/6*x^3 + 625/2*x^2 - 4475/3*x + 2240
5
7
4:
625/24*x^4 - 5875/12*x^3 + 78875/24*x^2 - 110675/12*x + 8960
5
7
5:
-625/24*x^5 + 6875/12*x^4 - 114875/24*x^3 + 224875/12*x^2 - 33700*x + 21504
5
7
6:
3125/144*x^6 - 25625/48*x^5 + 745625/144*x^4 - 400125/16*x^3 + 4472125/72*x^2 - 434945/6*x + 28672
5
7
7:
-15625/1008*x^7 + 59375/144*x^6 - 638125/144*x^5 + 3543125/144*x^4 - 2686375/36*x^3 + 4287425/36*x^2 - 1796005/21*x + 16384
5
8
0:
1
5
8
1:
-5*x + 32
5
8
2:
25/2*x^2 - 305/2*x + 448
5
8
3:
-125/6*x^3 + 725/2*x^2 - 6065/3*x + 3584
5
8
4:
625/24*x^4 - 6875/12*x^3 + 108875/24*x^2 - 182275/12*x + 17920
5
8
5:
-625/24*x^5 + 8125/12*x^4 - 161875/24*x^3 + 382625/12*x^2 - 211775/3*x + 57344
5
8
6:
3125/144*x^6 - 30625/48*x^5 + 1075625/144*x^4 - 2119375/48*x^3 + 9869125/72*x^2 - 1243745/6*x + 114688
5
8
7:
-15625/1008*x^7 + 71875/144*x^6 - 945625/144*x^5 + 6525625/144*x^4 - 1571875/9*x^3 + 13231675/36*x^2 - 7885235/21*x + 131072
5
8
8:
78125/8064*x^8 - 671875/2016*x^7 + 915625/192*x^6 - 5286875/144*x^5 + 20991875/128*x^4 - 122604625/288*x^3 + 1215617575/2016*x^2 - 65675285/168*x + 65536
5
9
0:
1
5
9
1:
-5*x + 36
5
9
2:
25/2*x^2 - 345/2*x + 576
5
9
3:
-125/6*x^3 + 825/2*x^2 - 7895/3*x + 5376
5
9
4:
625/24*x^4 - 2625/4*x^3 + 143675/24*x^2 - 93105/4*x + 32256
5
9
5:
-625/24*x^5 + 3125/4*x^4 - 216875/24*x^3 + 200125/4*x^2 - 131350*x + 129024
5
9
6:
3125/144*x^6 - 11875/16*x^5 + 1465625/144*x^4 - 1138125/16*x^3 + 19052125/72*x^2 - 979315/2*x + 344064
5
9
7:
-15625/1008*x^7 + 9375/16*x^6 - 1313125/144*x^5 + 1203125/16*x^4 - 12645625/36*x^3 + 3663325/4*x^2 - 8432555/7*x + 589824
5
9
8:
78125/8064*x^8 - 265625/672*x^7 + 3896875/576*x^6 - 3023125/48*x^5 + 397746875/1152*x^4 - 107934875/96*x^3 + 1393170925/672*x^2 - 106000935/56*x + 589824
5
9
9:
-390625/72576*x^9 + 78125/336*x^8 - 51671875/12096*x^7 + 521875/12*x^6 - 925118125/3456*x^5 + 48870625/48*x^4 - 42431912375/18144*x^3 + 63036800/21*x^2 - 224369605/126*x + 262144
5
10
0:
1
5
10
1:
-5*x + 40
5
10
2:
25/2*x^2 - 385/2*x + 720
5
10
3:
-125/6*x^3 + 925/2*x^2 - 9965/3*x + 7680
5
10
4:
625/24*x^4 - 8875/12*x^3 + 183275/24*x^2 - 405635/12*x + 53760
5
10
5:
-625/24*x^5 + 10625/12*x^4 - 279875/24*x^3 + 887725/12*x^2 - 224455*x + 258048
5
10
6:
3125/144*x^6 - 40625/48*x^5 + 1915625/144*x^4 - 5149375/48*x^3 + 33461125/72*x^2 - 2030115/2*x + 860160
5
10
7:
-15625/1008*x^7 + 96875/144*x^6 - 1740625/144*x^5 + 16690625/144*x^4 - 11444375/18*x^3 + 71074175/36*x^2 - 22142965/7*x + 1966080
5
10
8:
78125/8064*x^8 - 921875/2016*x^7 + 5246875/576*x^6 - 14321875/144*x^5 + 744246875/1152*x^4 - 728464625/288*x^3 + 3854925325/672*x^2 - 375842695/56*x + 2949120
5
10
9:
-390625/72576*x^9 + 78125/288*x^8 - 70796875/12096*x^7 + 10159375/144*x^6 - 1795778125/3456*x^5 + 690970625/288*x^4 - 124030677875/18144*x^3 + 271066475/24*x^2 - 589189010/63*x + 2621440
5
10
10:
390625/145152*x^10 - 20703125/145152*x^9 + 79140625/24192*x^8 - 1028609375/24192*x^7 + 2379353125/6912*x^6 - 12410710625/6912*x^5 + 27307080625/4536*x^4 - 453976783375/36288*x^3 + 14887831225/1008*x^2 - 1020525295/126*x + 1048576
5
11
0:
1
5
11
1:
-5*x + 44
5
11
2:
25/2*x^2 - 425/2*x + 880
5
11
3:
-125/6*x^3 + 1025/2*x^2 - 12275/3*x + 10560
5
11
4:
625/24*x^4 - 9875/12*x^3 + 227675/24*x^2 - 565075/12*x + 84480
5
11
5:
-625/24*x^5 + 11875/12*x^4 - 350875/24*x^3 + 1254275/12*x^2 - 1079000/3*x + 473088
5
11
6:
3125/144*x^6 - 45625/48*x^5 + 2425625/144*x^4 - 7388375/48*x^3 + 54766525/72*x^2 - 3825755/2*x + 1892352
5
11
7:
-15625/1008*x^7 + 109375/144*x^6 - 2228125/144*x^5 + 24353125/144*x^4 - 38336875/36*x^3 + 137996425/36*x^2 - 50564575/7*x + 5406720
5
11
8:
78125/8064*x^8 - 1046875/2016*x^7 + 2265625/192*x^6 - 21284375/144*x^5 + 426115625/384*x^4 - 1460904625/288*x^3 + 27485391175/2016*x^2 - 1084417575/56*x + 10813440
5
11
9:
-390625/72576*x^9 + 78125/252*x^8 - 92921875/12096*x^7 + 7703125/72*x^6 - 3170678125/3456*x^5 + 358804375/72*x^4 - 307603763375/18144*x^3 + 958731775/28*x^2 - 4560962275/126*x + 14417920
5
11
10:
390625/145152*x^10 - 23828125/145152*x^9 + 105390625/24192*x^8 - 1594984375/24192*x^7 + 4329953125/6912*x^6 - 26776935625/6912*x^5 + 8854778750/567*x^4 - 1446222206375/36288*x^3 + 60426999025/1008*x^2 - 5734037375/126*x + 11534336
5
11
11:
-1953125/1596672*x^11 + 11328125/145152*x^10 - 11328125/5184*x^9 + 121953125/3456*x^8 - 17566765625/48384*x^7 + 17144365625/6912*x^6 - 824599455625/72576*x^5 + 1247117595625/36288*x^4 - 170144873375/2592*x^3 + 5189770975/72*x^2 - 25527728365/693*x + 4194304
5
12
0:
1
5
12
1:
-5*x + 48
5
12
2:
25/2*x^2 - 465/2*x + 1056
5
12
3:
-125/6*x^3 + 1125/2*x^2 - 14825/3*x + 14080
5
12
4:
625/24*x^4 - 3625/4*x^3 + 276875/24*x^2 - 253825/4*x + 126720
5
12
5:
-625/24*x^5 + 4375/4*x^4 - 429875/24*x^3 + 569875/4*x^2 - 548025*x + 811008
5
12
6:
3125/144*x^6 - 16875/16*x^5 + 2995625/144*x^4 - 10195375/48*x^3 + 84869125/72*x^2 - 20109265/6*x + 3784704
5
12
7:
-15625/1008*x^7 + 40625/48*x^6 - 2775625/144*x^5 + 11351875/48*x^4 - 15125500/9*x^3 + 27503275/4*x^2 - 104125145/7*x + 12976128
5
12
8:
78125/8064*x^8 - 390625/672*x^7 + 8546875/576*x^6 - 10065625/48*x^5 + 2057646875/1152*x^4 - 298631625/32*x^3 + 19465530125/672*x^2 - 2702483975/56*x + 32440320
5
12
9:
-390625/72576*x^9 + 78125/224*x^8 - 118046875/12096*x^7 + 22203125/144*x^6 - 5213978125/3456*x^5 + 2713564375/288*x^4 - 675751728875/18144*x^3 + 44742563125/504*x^2 - 7160360225/63*x + 57671680
5
12
10:
390625/145152*x^10 - 8984375/48384*x^9 + 135390625/24192*x^8 - 779453125/8064*x^7 + 7287953125/6912*x^6 - 5793595625/768*x^5 + 40314233125/1134*x^4 - 1302350771125/12096*x^3 + 198484374625/1008*x^2 - 7992628825/42*x + 69206016
5
12
11:
-1953125/1596672*x^11 + 4296875/48384*x^10 - 4296875/1512*x^9 + 425078125/8064*x^8 - 30326640625/48384*x^7 + 3829353125/768*x^6 - 1949230751875/72576*x^5 + 1171313651875/12096*x^4 - 1361152842125/6048*x^3 + 52394131625/168*x^2 - 50558850205/231*x + 50331648
5
12
12:
9765625/19160064*x^12 - 1953125/50688*x^11 + 2252734375/1741824*x^10 - 7371484375/290304*x^9 + 187723984375/580608*x^8 - 270622015625/96768*x^7 + 29124147503125/1741824*x^6 - 952347818125/13824*x^5 + 83223255918125/435456*x^4 - 24638833239875/72576*x^3 + 23268844695475/66528*x^2 - 464840138215/2772*x + 16777216
5
13
0:
1
5
13
1:
-5*x + 52
5
13
2:
25/2*x^2 - 505/2*x + 1248
5
13
3:
-125/6*x^3 + 1225/2*x^2 - 17615/3*x + 18304
5
13
4:
625/24*x^4 - 11875/12*x^3 + 330875/24*x^2 - 998675/12*x + 183040
5
13
5:
-625/24*x^5 + 14375/12*x^4 - 516875/24*x^3 + 2263375/12*x^2 - 801850*x + 1317888
5
13
6:
3125/144*x^6 - 55625/48*x^5 + 3625625/144*x^4 - 13634375/48*x^3 + 125900125/72*x^2 - 33261865/6*x + 7028736
5
13
7:
-15625/1008*x^7 + 134375/144*x^6 - 3383125/144*x^5 + 46038125/144*x^4 - 91088125/36*x^3 + 417267725/36*x^2 - 593905145/21*x + 28114944
5
13
8:
78125/8064*x^8 - 1296875/2016*x^7 + 10496875/576*x^6 - 41299375/144*x^5 + 3147426875/1152*x^4 - 4623748625/288*x^3 + 37947730925/672*x^2 - 6034488615/56*x + 84344832
5
13
9:
-390625/72576*x^9 + 390625/1008*x^8 - 146171875/12096*x^7 + 640625/3*x^6 - 8112878125/3456*x^5 + 795201875/48*x^4 - 1353048254375/18144*x^3 + 25784788375/126*x^2 - 38643076225/126*x + 187432960
5
13
10:
390625/145152*x^10 - 4296875/20736*x^9 + 169140625/24192*x^8 - 3282734375/24192*x^7 + 11550953125/6912*x^6 - 93854185625/6912*x^5 + 332211488125/4536*x^4 - 1330438020625/5184*x^3 + 61824986625/112*x^2 - 81260768275/126*x + 299892736
5
13
11:
-1953125/1596672*x^11 + 14453125/145152*x^10 - 14453125/4032*x^9 + 1816796875/24192*x^8 - 49033515625/48384*x^7 + 63615990625/6912*x^6 - 4139209898125/72576*x^5 + 8674162795625/36288*x^4 - 3965854384375/6048*x^3 + 554151144125/504*x^2 - 226396684355/231*x + 327155712
5
13
12:
9765625/19160064*x^12 - 138671875/3193344*x^11 + 2871484375/1741824*x^10 - 10671484375/290304*x^9 + 44306640625/82944*x^8 - 513235140625/96768*x^7 + 63864039053125/1741824*x^6 - 51186996210625/290304*x^5 + 251892421788125/435456*x^4 - 89974169661875/72576*x^3 + 15180164169925/9504*x^2 - 2891664948055/2772*x + 218103808
5
13
13:
-48828125/249080832*x^13 + 166015625/9580032*x^12 - 13158203125/19160064*x^11 + 14026953125/870912*x^10 - 144274609375/580608*x^9 + 109948203125/41472*x^8 - 34846943328125/1741824*x^7 + 93430179771875/870912*x^6 - 176097534068125/435456*x^5 + 226799392695625/217728*x^4 - 346599626576125/199584*x^3 + 2008234303025/1188*x^2 - 6888869062420/9009*x + 67108864
5
14
0:
1
5
14
1:
-5*x + 56
5
14
2:
25/2*x^2 - 545/2*x + 1456
5
14
3:
-125/6*x^3 + 1325/2*x^2 - 20645/3*x + 23296
5
14
4:
625/24*x^4 - 12875/12*x^3 + 389675/24*x^2 - 1280515/12*x + 256256
5
14
5:
-625/24*x^5 + 15625/12*x^4 - 611875/24*x^3 + 2925125/12*x^2 - 3404225/3*x + 2050048
5
14
6:
3125/144*x^6 - 60625/48*x^5 + 4315625/144*x^4 - 5923125/16*x^3 + 180221125/72*x^2 - 52506265/6*x + 12300288
5
14
7:
-15625/1008*x^7 + 146875/144*x^6 - 4050625/144*x^5 + 60540625/144*x^4 - 65995625/18*x^3 + 669067975/36*x^2 - 1059571255/21*x + 56229888
5
14
8:
78125/8064*x^8 - 203125/288*x^7 + 4215625/192*x^6 - 54831875/144*x^5 + 1540215625/384*x^4 - 7538568625/288*x^3 + 207311163175/2016*x^2 - 5301204355/24*x + 196804608
5
14
9:
-390625/72576*x^9 + 859375/2016*x^8 - 25328125/1728*x^7 + 41246875/144*x^6 - 12077618125/3456*x^5 + 7918638125/288*x^4 - 2518232907875/18144*x^3 + 216982346275/504*x^2 - 6639533840/9*x + 524812288
5
14
10:
390625/145152*x^10 - 33203125/145152*x^9 + 206640625/24192*x^8 - 636015625/3456*x^7 + 17454953125/6912*x^6 - 158757210625/6912*x^5 + 632797796875/4536*x^4 - 20137452179375/36288*x^3 + 1381538107625/1008*x^2 - 33690439025/18*x + 1049624576
5
14
11:
-1953125/1596672*x^11 + 16015625/145152*x^10 - 80078125/18144*x^9 + 2493359375/24192*x^8 - 10756484375/6912*x^7 + 109819803125/6912*x^6 - 8081085694375/72576*x^5 + 19304930415625/36288*x^4 - 30523695441875/18144*x^3 + 1667000903375/504*x^2 - 352418136445/99*x + 1526726656
5
14
12:
9765625/19160064*x^12 - 154296875/3193344*x^11 + 3565234375/1741824*x^10 - 2119140625/41472*x^9 + 484558984375/580608*x^8 - 43119203125/4608*x^7 + 127988957603125/1741824*x^6 - 117414354580625/290304*x^5 + 668252235978125/435456*x^4 - 40047882873125/10368*x^3 + 398852953287475/66528*x^2 - 1965529399585/396*x + 1526726656
5
14
13:
-48828125/249080832*x^13 + 185546875/9580032*x^12 - 16486328125/19160064*x^11 + 19769921875/870912*x^10 - 10935546875/27648*x^9 + 198561484375/41472*x^8 - 71799873453125/1741824*x^7 + 221158257878125/870912*x^6 - 483219511331875/435456*x^5 + 730584236271875/217728*x^4 - 95451106632625/14256*x^3 + 19196632775975/2376*x^2 - 6354359055305/1287*x + 939524096
5
14
14:
244140625/3487131648*x^14 - 3564453125/498161664*x^13 + 12666015625/38320128*x^12 - 349650390625/38320128*x^11 + 584050390625/3483648*x^10 - 278824609375/129024*x^9 + 489142851484375/24385536*x^8 - 471683993890625/3483648*x^7 + 1154275666690625/1741824*x^6 - 2015943114341875/870912*x^5 + 26770726195886875/4790016*x^4 - 439510271107375/49896*x^3 + 24664675725415175/3027024*x^2 - 62966177108735/18018*x + 268435456
5
15
0:
1
5
15
1:
-5*x + 60
5
15
2:
25/2*x^2 - 585/2*x + 1680
5
15
3:
-125/6*x^3 + 1425/2*x^2 - 23915/3*x + 29120
5
15
4:
625/24*x^4 - 4625/4*x^3 + 453275/24*x^2 - 536945/4*x + 349440
5
15
5:
-625/24*x^5 + 5625/4*x^4 - 714875/24*x^3 + 1234825/4*x^2 - 1561580*x + 3075072
5
15
6:
3125/144*x^6 - 21875/16*x^5 + 5065625/144*x^4 - 22664375/48*x^3 + 250424125/72*x^2 - 79740065/6*x + 20500480
5
15
7:
-15625/1008*x^7 + 53125/48*x^6 - 4778125/144*x^5 + 25934375/48*x^4 - 185299375/36*x^3 + 114390025/4*x^2 - 598219655/7*x + 105431040
5
15
8:
78125/8064*x^8 - 171875/224*x^7 + 14996875/576*x^6 - 23678125/48*x^5 + 6557946875/1152*x^4 - 3920762875/96*x^3 + 39686932175/224*x^2 - 23671570215/56*x + 421724160
5
15
9:
-390625/72576*x^9 + 78125/168*x^8 - 30203125/1728*x^7 + 26946875/72*x^6 - 17341478125/3456*x^5 + 1567410625/36*x^4 - 4417952201375/18144*x^3 + 212146754725/252*x^2 - 29182680745/18*x + 1312030720
5
15
10:
390625/145152*x^10 - 12109375/48384*x^9 + 247890625/24192*x^8 - 279546875/1152*x^7 + 25374353125/6912*x^6 - 28375350625/768*x^5 + 565835999375/2268*x^4 - 13427771814125/12096*x^3 + 3117396877825/1008*x^2 - 28935569915/6*x + 3148873728
5
15
11:
-1953125/1596672*x^11 + 1953125/16128*x^10 - 21484375/4032*x^9 + 1106640625/8064*x^8 - 15844609375/6912*x^7 + 59879871875/2304*x^6 - 14748888540625/72576*x^5 + 488326665625/448*x^4 - 23599533266875/6048*x^3 + 1476692372875/168*x^2 - 364535931665/33*x + 5725224960
5
15
12:
9765625/19160064*x^12 - 56640625/1064448*x^11 + 619140625/248832*x^10 - 19958984375/290304*x^9 + 723921484375/580608*x^8 - 215409484375/13824*x^7 + 238687319153125/1741824*x^6 - 9137471321875/10752*x^5 + 227841270846875/62208*x^4 - 768714307181875/72576*x^3 + 1279029430269475/66528*x^2 - 7604219582705/396*x + 7633633280
5
15
13:
-48828125/249080832*x^13 + 9765625/456192*x^12 - 20189453125/19160064*x^11 + 8966796875/290304*x^10 - 49759765625/82944*x^9 + 262116484375/32256*x^8 - 136996108578125/1741824*x^7 + 22720770146875/41472*x^6 - 1187705638815625/435456*x^5 + 689029569409375/72576*x^4 - 210476308289875/9504*x^3 + 44435781893275/1386*x^2 - 10635413749970/429*x + 7046430720
5
15
14:
244140625/3487131648*x^14 - 48828125/6150144*x^13 + 15634765625/38320128*x^12 - 160513671875/12773376*x^11 + 100041015625/387072*x^10 - 1448864453125/387072*x^9 + 956159462734375/24385536*x^8 - 116231442390625/387072*x^7 + 2923541729715625/1741824*x^6 - 1960566401665625/290304*x^5 + 30354046329270625/1596672*x^4 - 591941921321375/16632*x^3 + 40830238783927925/1009008*x^2 - 139603428068605/6006*x + 4026531840
5
15
15:
-244140625/10461394944*x^15 + 244140625/89413632*x^14 - 217333984375/1494484992*x^13 + 178837890625/38320128*x^12 - 11573001953125/114960384*x^11 + 5369648828125/3483648*x^10 - 1261354926953125/73156608*x^9 + 1160855426328125/8128512*x^8 - 2293648661359375/2612736*x^7 + 216263675865625/54432*x^6 - 23327020500449375/1796256*x^5 + 70923575194508125/2395008*x^4 - 403072336497028375/9081072*x^3 + 4564747568783825/116424*x^2 - 144261211639345/9009*x + 1073741824
5
16
0:
1
5
16
1:
-5*x + 64
5
16
2:
25/2*x^2 - 625/2*x + 1920
5
16
3:
-125/6*x^3 + 1525/2*x^2 - 27425/3*x + 35840
5
16
4:
625/24*x^4 - 14875/12*x^3 + 521675/24*x^2 - 1993475/12*x + 465920
5
16
5:
-625/24*x^5 + 18125/12*x^4 - 825875/24*x^3 + 4611025/12*x^2 - 2098525*x + 4472832
5
16
6:
3125/144*x^6 - 70625/48*x^5 + 5875625/144*x^4 - 9461125/16*x^3 + 339331525/72*x^2 - 117217985/6*x + 32800768
5
16
7:
-15625/1008*x^7 + 171875/144*x^6 - 5565625/144*x^5 + 98065625/144*x^4 - 63323125/9*x^3 + 1530358475/36*x^2 - 2911019875/21*x + 187432960
5
16
8:
78125/8064*x^8 - 1671875/2016*x^7 + 17546875/576*x^6 - 90146875/144*x^5 + 9047646875/1152*x^4 - 17691868625/288*x^3 + 65310297775/224*x^2 - 42814599175/56*x + 843448320
5
16
9:
-390625/72576*x^9 + 1015625/2016*x^8 - 248546875/12096*x^7 + 68890625/144*x^6 - 24160778125/3456*x^5 + 19097231875/288*x^4 - 7382048934875/18144*x^3 + 781475899025/504*x^2 - 208661448575/63*x + 2998927360
5
16
10:
390625/145152*x^10 - 39453125/145152*x^9 + 292890625/24192*x^8 - 1080265625/3456*x^7 + 35721953125/6912*x^6 - 394109980625/6912*x^5 + 960823476875/2268*x^4 - 75626933053375/36288*x^3 + 6511744953425/1008*x^2 - 203537432725/18*x + 8396996608
5
16
11:
-1953125/1596672*x^11 + 2734375/20736*x^10 - 2734375/432*x^9 + 4311484375/24192*x^8 - 22553734375/6912*x^7 + 281137028125/6912*x^6 - 25474771076875/72576*x^5 + 10823994839375/5184*x^4 - 7207868127875/864*x^3 + 10664870874275/504*x^2 - 1001118469795/33*x + 18320719872
5
16
12:
9765625/19160064*x^12 - 185546875/3193344*x^11 + 5177734375/1741824*x^10 - 26146484375/290304*x^9 + 1042633984375/580608*x^8 - 12672828125/512*x^7 + 419764051703125/1741824*x^6 - 482693942340625/290304*x^5 + 3493502971878125/435456*x^4 - 1901491903991875/72576*x^3 + 3618110148903475/66528*x^2 - 25101944302625/396*x + 30534533120
5
16
13:
-48828125/249080832*x^13 + 224609375/9580032*x^12 - 3466796875/2737152*x^11 + 35568359375/870912*x^10 - 6271484375/7168*x^9 + 3806891328125/290304*x^8 - 245562488703125/1741824*x^7 + 954510811390625/870912*x^6 - 381139427565625/62208*x^5 + 5256866500084375/217728*x^4 - 402370620409000/6237*x^3 + 1812258812988775/16632*x^2 - 130761095825075/1287*x + 37580963840
5
16
14:
244140625/3487131648*x^14 - 4345703125/498161664*x^13 + 18916015625/38320128*x^12 - 91865234375/5474304*x^11 + 443591796875/1161216*x^10 - 7133140234375/1161216*x^9 + 1748799711484375/24385536*x^8 - 2142051850140625/3483648*x^7 + 6740631114390625/1741824*x^6 - 2197620616503125/124416*x^5 + 90988648437295625/1596672*x^4 - 172106339945875/1386*x^3 + 170227235657144725/1009008*x^2 - 735186598066925/6006*x + 32212254720
5
16
15:
-244140625/10461394944*x^15 + 10498046875/3487131648*x^14 - 24072265625/135862272*x^13 + 1044921875/165888*x^12 - 17351494140625/114960384*x^11 + 8971125390625/3483648*x^10 - 2356696453515625/73156608*x^9 + 7307204129921875/24385536*x^8 - 2715948802953125/1306368*x^7 + 73867478359375/6912*x^6 - 287404155766694375/7185024*x^5 + 253047853170131875/2395008*x^4 - 154170317514810125/825552*x^3 + 27665741008887025/137592*x^2 - 981881780050975/9009*x + 17179869184
5
16
16:
1220703125/167382319104*x^16 - 20263671875/20922789888*x^15 + 2465087890625/41845579776*x^14 - 1624853515625/747242496*x^13 + 4548271484375/83607552*x^12 - 112155044921875/114960384*x^11 + 3784847758203125/292626432*x^10 - 4712193496484375/36578304*x^9 + 1135005107156796875/1170505728*x^8 - 114981638755296875/20901888*x^7 + 5346999166790834375/229920768*x^6 - 2060274008073681875/28740096*x^5 + 541664953157180554375/3487131648*x^4 - 32330164978925050375/145297152*x^3 + 414926703474042775/2201472*x^2 - 10607381851558705/144144*x + 4294967296
Definition
For an alphabet size $q\geq2$, length $N\geq1$ and degree $0\leq k\leq N$, $\mathcal{K}_k(x;N,q)$ is the Hamming-scheme Krawtchouk polynomial $\sum_{j=0}^{k}(-1)^j(q-1)^{k-j}\binom{x}{j}\binom{N-x}{k-j}$, with $\binom{x}{j}$ read as a falling-factorial polynomial in $x$.
Parameters
$q$
—   alphabet size ($q\geq 2$)
$N$
—   length ($N\geq 1$)
$k$
—   degree ($0\leq k\leq N$)
Formulas
(1)
$\sum_{k=0}^{N}\mathcal{K}_k(x;N,q)z^k=(1+(q-1)z)^{N-x}(1-z)^x$.
(2)
$\sum_{x=0}^{N}\binom{N}{x}(q-1)^x\mathcal{K}_k(x;N,q)\mathcal{K}_l(x;N,q)=\delta_{kl}q^N\binom{N}{k}(q-1)^k$.
Comments
(3)
$\binom{x}{j}$ is $x(x-1)\cdots(x-j+1)/j!$, so $\mathcal{K}_k$ is a degree-$k$ polynomial in $x$. At $x=0,1,\ldots,N$, $x$ records the Hamming weight [2].
(4)
This is the Hamming-scheme normalisation. When $q$ is a prime power, it is the normalisation used for the MacWilliams identity [1]. If $K_k(x;p,N)$ denotes the Askey-scheme normalisation of [3], then $\mathcal{K}_k(x;N,q)=\binom{N}{k}(q-1)^kK_k(x;1-1/q,N)$.
(5)
For the Hamming scheme $H(N,q)$, the value $\mathcal{K}_k(i;N,q)$ is the eigenvalue of the distance-$k$ adjacency matrix on the $i$-th eigenspace.
Programs
(P1)
Sage
def krawtchouk_hamming(q, N, k):
    from sage.arith.misc import binomial
    from sage.rings.rational_field import QQ
    from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
    R = PolynomialRing(QQ, "x")
    x = R.gen()
    return sum((-1)**j * (q - 1)**(k - j) * binomial(x, j) * binomial(N - x, k - j)
               for j in range(k + 1))
References
[1]
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, 1977.
Links
[2]
https://en.wikipedia.org/wiki/Kravchuk_polynomials
[3]
https://dlmf.nist.gov/18.19
[4]
https://mathworld.wolfram.com/KrawtchoukPolynomial.html
Similar tables
Gegenbauer polynomials —   they play the same part on the sphere that $\mathcal{K}_k$ plays in Hamming space, as the polynomials in Delsarte's linear programming bound
Kissing numbers $\tau_n$ —   $\tau_8=240$ and $\tau_{24}=196560$ were proved with Delsarte's linear programming bound, whose Hamming-space form is written with these polynomials
Data properties
Entries are of type: rational polynomial
How well the digits are known: exact
Table is complete: no (it holds $q\in\{2,3,4,5\}$, every length $1\leq N\leq 16$ and every degree $0\leq k\leq N$, the small alphabet sizes and short code lengths most often written out)
How they were obtained:

Each polynomial is computed exactly over $\mathbb{Q}$ from the falling-factorial definition.

more

Before the draft was filled, the generator was checked against the generating function, the orthogonality relation on the Hamming scheme, the relation with the Askey-scheme normalisation, and the first four binary polynomials printed in the article [2].



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