History of Heat (caloric) polynomials $P_m(x,t)$

 Title: Heat (caloric) polynomials $P_m(x,t)$-Definition: The heat polynomials $P_m(x,t)$ are the parabolically $m$-homogeneous-  polynomial solutions of $\partial P/\partial t=\partial^2P/\partial x^2$, normalized-  by $P_m(x,0)=x^m$ CITE{Wiki}.+Definition: The heat polynomial $P_m(x,t)$ is the unique polynomial solution of $\partial+  P/\partial t=\partial^2P/\partial x^2$ with $P_m(x,0)=x^m$ CITE{RW}. Parameters:   m:

 Formulas:   formula-explicit: $P_m(x,t)=\sum_{\ell=0}^{\lfloor m/2\rfloor} \frac{m!}{\ell!(m-2\ell)!}x^{m-2\ell}t^\ell$.+  formula-homogeneous: $P_m(\lambda x,\lambda^2t)=\lambda^mP_m(x,t)$ for $\lambda>0$;+    this is parabolic $m$-homogeneity.   formula-recurrence: $P_0(x,t)=1$ and $P_{m+1}(x,t)=xP_m(x,t)+2t\,\partial P_m(x,t)/\partial     x$.

   program-sage:     language: Sage-    code: "import numberdb.sage as numberdb\nfrom sage.rings.integer_ring import ZZ\n\-      from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing\n\-      \nR = PolynomialRing(ZZ, ('x', 't'))\nx, t = R.gens()\n\ndef heat_polynomial(m):\n\-      \    p = R.one()\n    for _ in range(m):\n        p = x * p + ZZ(2) * t * p.derivative(x)\n\-      \    return p\n\nheat_polynomial(51)      # the next one after this table"+    code: "from sage.all import ZZ, PolynomialRing\n\nR = PolynomialRing(ZZ, ('x',\+      \ 't'))\nx, t = R.gens()\n\ndef heat_polynomial(m):\n    p = R.one()\n    for\+      \ _ in range(m):\n        p = x * p + ZZ(2) * t * p.derivative(x)\n    return\+      \ p\n\nheat_polynomial(51)      # the next one after this table" Similar tables: - table: HREF{Hermite_polynomials_He}[Hermite polynomials in probabilist's convention]

     title: 'Wikipedia: Caloric polynomial'     url: https://en.wikipedia.org/wiki/Caloric_polynomial+References:+  RW:+    bib: P. C. Rosenbloom and D. V. Widder, Expansions in terms of heat polynomials+      and associated functions, Trans. Amer. Math. Soc. 92 (1959), 220-266.+    doi: 10.2307/1993155+    zbl: '0086.27203' Keywords:-- caloric polynomial-- caloric polynomials-- heat polynomial+- heat equation+- parabolic homogeneity+- parabolically homogeneous+- Rosenbloom+- Widder Tags: - polynomial

   rigour: exact   complete: 'no'-  complete-note: it holds every degree $m$ with $0\leq m\leq50$; $P_{50}$ is 1016-    characters, and $P_{60}$ is 1425 characters-  rigour details: Every entry is exact, a polynomial with integer coefficients. The-    generator computes CITE{formula-recurrence} over Sage's integer polynomial ring.-    Before the entries were sent, the dry run computed all 51 entries and found exact-    coefficients; the values were checked against CITE{formula-explicit}, against-    the heat equation, against the normalization $P_m(x,0)=x^m$, against parabolic-    homogeneity, and against the specialization to HREF{Hermite_polynomials_He}[probabilists'-    Hermite polynomials] in CITE{formula-hermite}, read from T104.+  complete-note: 'it holds every degree $m$ with $0\leq m\leq50$, and stops there+    because the entries grow: $P_{50}$ is 1016 characters, and $P_{60}$ would be 1425'+  rigour details: The generator computes CITE{formula-recurrence} over Sage's integer+    polynomial ring. The values were checked against CITE{formula-explicit}, against+    the heat equation, against the normalization $P_m(x,0)=x^m$, against the parabolic+    homogeneity identity CITE{formula-homogeneous}, and against the specialization+    to HREF{Hermite_polynomials_He}[probabilists' Hermite polynomials] in CITE{formula-hermite}. Display properties:   number-header: $P_m(x,t)$ Numbers:-- params:-    m: '0'-  number: '1'-- params:-    m: '1'-  number: x-- params:-    m: '2'-  number: x^2 + 2*t-- params:-    m: '3'-  number: x^3 + 6*x*t-- params:-    m: '4'-  number: x^4 + 12*x^2*t + 12*t^2-- params:-    m: '5'-  number: x^5 + 20*x^3*t + 60*x*t^2-- params:-    m: '6'-  number: x^6 + 30*x^4*t + 180*x^2*t^2 + 120*t^3-- params:-    m: '7'-  number: x^7 + 42*x^5*t + 420*x^3*t^2 + 840*x*t^3-- params:-    m: '8'-  number: x^8 + 56*x^6*t + 840*x^4*t^2 + 3360*x^2*t^3 + 1680*t^4-- params:-    m: '9'-  number: x^9 + 72*x^7*t + 1512*x^5*t^2 + 10080*x^3*t^3 + 15120*x*t^4-- params:-    m: '10'-  number: x^10 + 90*x^8*t + 2520*x^6*t^2 + 25200*x^4*t^3 + 75600*x^2*t^4 + 30240*t^5-- params:-    m: '11'-  number: x^11 + 110*x^9*t + 3960*x^7*t^2 + 55440*x^5*t^3 + 277200*x^3*t^4 + 332640*x*t^5-- params:-    m: '12'-  number: x^12 + 132*x^10*t + 5940*x^8*t^2 + 110880*x^6*t^3 + 831600*x^4*t^4 + 1995840*x^2*t^5+  '0': '1'+  '1': x+  '2': x^2 + 2*t+  '3': x^3 + 6*x*t+  '4': x^4 + 12*x^2*t + 12*t^2+  '5': x^5 + 20*x^3*t + 60*x*t^2+  '6': x^6 + 30*x^4*t + 180*x^2*t^2 + 120*t^3+  '7': x^7 + 42*x^5*t + 420*x^3*t^2 + 840*x*t^3+  '8': x^8 + 56*x^6*t + 840*x^4*t^2 + 3360*x^2*t^3 + 1680*t^4+  '9': x^9 + 72*x^7*t + 1512*x^5*t^2 + 10080*x^3*t^3 + 15120*x*t^4+  '10': x^10 + 90*x^8*t + 2520*x^6*t^2 + 25200*x^4*t^3 + 75600*x^2*t^4 + 30240*t^5+  '11': x^11 + 110*x^9*t + 3960*x^7*t^2 + 55440*x^5*t^3 + 277200*x^3*t^4 + 332640*x*t^5+  '12': x^12 + 132*x^10*t + 5940*x^8*t^2 + 110880*x^6*t^3 + 831600*x^4*t^4 + 1995840*x^2*t^5     + 665280*t^6-- params:-    m: '13'-  number: x^13 + 156*x^11*t + 8580*x^9*t^2 + 205920*x^7*t^3 + 2162160*x^5*t^4 + 8648640*x^3*t^5+  '13': x^13 + 156*x^11*t + 8580*x^9*t^2 + 205920*x^7*t^3 + 2162160*x^5*t^4 + 8648640*x^3*t^5     + 8648640*x*t^6-- params:-    m: '14'-  number: x^14 + 182*x^12*t + 12012*x^10*t^2 + 360360*x^8*t^3 + 5045040*x^6*t^4 +-    30270240*x^4*t^5 + 60540480*x^2*t^6 + 17297280*t^7-- params:-    m: '15'-  number: x^15 + 210*x^13*t + 16380*x^11*t^2 + 600600*x^9*t^3 + 10810800*x^7*t^4 +-    90810720*x^5*t^5 + 302702400*x^3*t^6 + 259459200*x*t^7-- params:-    m: '16'-  number: x^16 + 240*x^14*t + 21840*x^12*t^2 + 960960*x^10*t^3 + 21621600*x^8*t^4-    + 242161920*x^6*t^5 + 1210809600*x^4*t^6 + 2075673600*x^2*t^7 + 518918400*t^8-- params:-    m: '17'-  number: x^17 + 272*x^15*t + 28560*x^13*t^2 + 1485120*x^11*t^3 + 40840800*x^9*t^4-    + 588107520*x^7*t^5 + 4116752640*x^5*t^6 + 11762150400*x^3*t^7 + 8821612800*x*t^8-- params:-    m: '18'-  number: x^18 + 306*x^16*t + 36720*x^14*t^2 + 2227680*x^12*t^3 + 73513440*x^10*t^4+  '14': x^14 + 182*x^12*t + 12012*x^10*t^2 + 360360*x^8*t^3 + 5045040*x^6*t^4 + 30270240*x^4*t^5+    + 60540480*x^2*t^6 + 17297280*t^7+  '15': x^15 + 210*x^13*t + 16380*x^11*t^2 + 600600*x^9*t^3 + 10810800*x^7*t^4 + 90810720*x^5*t^5+    + 302702400*x^3*t^6 + 259459200*x*t^7+  '16': x^16 + 240*x^14*t + 21840*x^12*t^2 + 960960*x^10*t^3 + 21621600*x^8*t^4 ++    242161920*x^6*t^5 + 1210809600*x^4*t^6 + 2075673600*x^2*t^7 + 518918400*t^8+  '17': x^17 + 272*x^15*t + 28560*x^13*t^2 + 1485120*x^11*t^3 + 40840800*x^9*t^4 ++    588107520*x^7*t^5 + 4116752640*x^5*t^6 + 11762150400*x^3*t^7 + 8821612800*x*t^8+  '18': x^18 + 306*x^16*t + 36720*x^14*t^2 + 2227680*x^12*t^3 + 73513440*x^10*t^4     + 1323241920*x^8*t^5 + 12350257920*x^6*t^6 + 52929676800*x^4*t^7 + 79394515200*x^2*t^8     + 17643225600*t^9-- params:-    m: '19'-  number: x^19 + 342*x^17*t + 46512*x^15*t^2 + 3255840*x^13*t^3 + 126977760*x^11*t^4+  '19': x^19 + 342*x^17*t + 46512*x^15*t^2 + 3255840*x^13*t^3 + 126977760*x^11*t^4     + 2793510720*x^9*t^5 + 33522128640*x^7*t^6 + 201132771840*x^5*t^7 + 502831929600*x^3*t^8     + 335221286400*x*t^9-- params:-    m: '20'-  number: x^20 + 380*x^18*t + 58140*x^16*t^2 + 4651200*x^14*t^3 + 211629600*x^12*t^4+  '20': x^20 + 380*x^18*t + 58140*x^16*t^2 + 4651200*x^14*t^3 + 211629600*x^12*t^4     + 5587021440*x^10*t^5 + 83805321600*x^8*t^6 + 670442572800*x^6*t^7 + 2514159648000*x^4*t^8     + 3352212864000*x^2*t^9 + 670442572800*t^10-- params:-    m: '21'-  number: x^21 + 420*x^19*t + 71820*x^17*t^2 + 6511680*x^15*t^3 + 341863200*x^13*t^4+  '21': x^21 + 420*x^19*t + 71820*x^17*t^2 + 6511680*x^15*t^3 + 341863200*x^13*t^4     + 10666131840*x^11*t^5 + 195545750400*x^9*t^6 + 2011327718400*x^7*t^7 + 10559470521600*x^5*t^8     + 23465490048000*x^3*t^9 + 14079294028800*x*t^10-- params:-    m: '22'-  number: x^22 + 462*x^20*t + 87780*x^18*t^2 + 8953560*x^16*t^3 + 537213600*x^14*t^4+  '22': x^22 + 462*x^20*t + 87780*x^18*t^2 + 8953560*x^16*t^3 + 537213600*x^14*t^4     + 19554575040*x^12*t^5 + 430200650880*x^10*t^6 + 5531151225600*x^8*t^7 + 38718058579200*x^6*t^8     + 129060195264000*x^4*t^9 + 154872234316800*x^2*t^10 + 28158588057600*t^11-- params:-    m: '23'-  number: x^23 + 506*x^21*t + 106260*x^19*t^2 + 12113640*x^17*t^3 + 823727520*x^15*t^4+  '23': x^23 + 506*x^21*t + 106260*x^19*t^2 + 12113640*x^17*t^3 + 823727520*x^15*t^4     + 34596555840*x^13*t^5 + 899510451840*x^11*t^6 + 14135164243200*x^9*t^7 + 127216478188800*x^7*t^8     + 593676898214400*x^5*t^9 + 1187353796428800*x^3*t^10 + 647647525324800*x*t^11-- params:-    m: '24'-  number: x^24 + 552*x^22*t + 127512*x^20*t^2 + 16151520*x^18*t^3 + 1235591280*x^16*t^4+  '24': x^24 + 552*x^22*t + 127512*x^20*t^2 + 16151520*x^18*t^3 + 1235591280*x^16*t^4     + 59308381440*x^14*t^5 + 1799020903680*x^12*t^6 + 33924394183680*x^10*t^7 + 381649434566400*x^8*t^8     + 2374707592857600*x^6*t^9 + 7124122778572800*x^4*t^10 + 7771770303897600*x^2*t^11     + 1295295050649600*t^12-- params:-    m: '25'-  number: x^25 + 600*x^23*t + 151800*x^21*t^2 + 21252000*x^19*t^3 + 1817046000*x^17*t^4+  '25': x^25 + 600*x^23*t + 151800*x^21*t^2 + 21252000*x^19*t^3 + 1817046000*x^17*t^4     + 98847302400*x^15*t^5 + 3459655584000*x^13*t^6 + 77100895872000*x^11*t^7 + 1060137318240000*x^9*t^8     + 8481098545920000*x^7*t^9 + 35620613892864000*x^5*t^10 + 64764752532480000*x^3*t^11     + 32382376266240000*x*t^12-- params:-    m: '26'-  number: x^26 + 650*x^24*t + 179400*x^22*t^2 + 27627600*x^20*t^3 + 2624622000*x^18*t^4+  '26': x^26 + 650*x^24*t + 179400*x^22*t^2 + 27627600*x^20*t^3 + 2624622000*x^18*t^4     + 160626866400*x^16*t^5 + 6425074656000*x^14*t^6 + 167051941056000*x^12*t^7 +     2756357027424000*x^10*t^8 + 27563570274240000*x^8*t^9 + 154355993535744000*x^6*t^10     + 420970891461120000*x^4*t^11 + 420970891461120000*x^2*t^12 + 64764752532480000*t^13-- params:-    m: '27'-  number: x^27 + 702*x^25*t + 210600*x^23*t^2 + 35521200*x^21*t^3 + 3729726000*x^19*t^4+  '27': x^27 + 702*x^25*t + 210600*x^23*t^2 + 35521200*x^21*t^3 + 3729726000*x^19*t^4     + 255113258400*x^17*t^5 + 11565134380800*x^15*t^6 + 346954031424000*x^13*t^7 +     6765603612768000*x^11*t^8 + 82690710822720000*x^9*t^9 + 595373117923584000*x^7*t^10     + 2273242813890048000*x^5*t^11 + 3788738023150080000*x^3*t^12 + 1748648318376960000*x*t^13-- params:-    m: '28'-  number: x^28 + 756*x^26*t + 245700*x^24*t^2 + 45208800*x^22*t^3 + 5221616400*x^20*t^4+  '28': x^28 + 756*x^26*t + 245700*x^24*t^2 + 45208800*x^22*t^3 + 5221616400*x^20*t^4     + 396842846400*x^18*t^5 + 20238985166400*x^16*t^6 + 693908062848000*x^14*t^7 +     15786408429792000*x^12*t^8 + 231533990303616000*x^10*t^9 + 2083805912732544000*x^8*t^10     + 10608466464820224000*x^6*t^11 + 26521166162050560000*x^4*t^12 + 24481076457277440000*x^2*t^13     + 3497296636753920000*t^14-- params:-    m: '29'-  number: x^29 + 812*x^27*t + 285012*x^25*t^2 + 57002400*x^23*t^3 + 7210803600*x^21*t^4+  '29': x^29 + 812*x^27*t + 285012*x^25*t^2 + 57002400*x^23*t^3 + 7210803600*x^21*t^4     + 605707502400*x^19*t^5 + 34525327636800*x^17*t^6 + 1341555588172800*x^15*t^7     + 35215834189536000*x^13*t^8 + 610407792618624000*x^11*t^9 + 6714485718804864000*x^9*t^10     + 43949361068540928000*x^7*t^11 + 153822763739893248000*x^5*t^12 + 236650405753681920000*x^3*t^13     + 101421602465863680000*x*t^14-- params:-    m: '30'-  number: x^30 + 870*x^28*t + 328860*x^26*t^2 + 71253000*x^24*t^3 + 9832914000*x^22*t^4+  '30': x^30 + 870*x^28*t + 328860*x^26*t^2 + 71253000*x^24*t^3 + 9832914000*x^22*t^4     + 908561253600*x^20*t^5 + 57542212728000*x^18*t^6 + 2515416727824000*x^16*t^7     + 75462501834720000*x^14*t^8 + 1526019481546560000*x^12*t^9 + 20143457156414592000*x^10*t^10     + 164810104007028480000*x^8*t^11 + 769113818699466240000*x^6*t^12 + 1774878043152614400000*x^4*t^13     + 1521324036987955200000*x^2*t^14 + 202843204931727360000*t^15-- params:-    m: '31'-  number: x^31 + 930*x^29*t + 377580*x^27*t^2 + 88353720*x^25*t^3 + 13253058000*x^23*t^4+  '31': x^31 + 930*x^29*t + 377580*x^27*t^2 + 88353720*x^25*t^3 + 13253058000*x^23*t^4     + 1341209469600*x^21*t^5 + 93884662872000*x^19*t^6 + 4586936386032000*x^17*t^7     + 155955837125088000*x^15*t^8 + 3638969532918720000*x^13*t^9 + 56767924713532032000*x^11*t^10     + 567679247135320320000*x^9*t^11 + 3406075482811921920000*x^7*t^12 + 11004243867546209280000*x^5*t^13     + 15720348382208870400000*x^3*t^14 + 6288139352883548160000*x*t^15-- params:-    m: '32'-  number: x^32 + 992*x^30*t + 431520*x^28*t^2 + 108743040*x^26*t^3 + 17670744000*x^24*t^4+  '32': x^32 + 992*x^30*t + 431520*x^28*t^2 + 108743040*x^26*t^3 + 17670744000*x^24*t^4     + 1950850137600*x^22*t^5 + 150215460595200*x^20*t^6 + 8154553575168000*x^18*t^7     + 311911674250176000*x^16*t^8 + 8317644646671360000*x^14*t^9 + 151381132569418752000*x^12*t^10     + 1816573590833025024000*x^10*t^11 + 13624301931247687680000*x^8*t^12 + 58689300626913116160000*x^6*t^13     + 125762787057670963200000*x^4*t^14 + 100610229646136770560000*x^2*t^15 + 12576278705767096320000*t^16-- params:-    m: '33'-  number: x^33 + 1056*x^31*t + 491040*x^29*t^2 + 132908160*x^27*t^3 + 23325382080*x^25*t^4+  '33': x^33 + 1056*x^31*t + 491040*x^29*t^2 + 132908160*x^27*t^3 + 23325382080*x^25*t^4     + 2799045849600*x^23*t^5 + 236052866649600*x^21*t^6 + 14163171998976000*x^19*t^7     + 605475602956224000*x^17*t^8 + 18298818222676992000*x^15*t^9 + 384275182676216832000*x^13*t^10     + 5449720772499075072000*x^11*t^11 + 49955773747908188160000*x^9*t^12 + 276678131526876119040000*x^7*t^13     + 830034394580628357120000*x^5*t^14 + 1106712526107504476160000*x^3*t^15 + 415017197290314178560000*x*t^16-- params:-    m: '34'-  number: x^34 + 1122*x^32*t + 556512*x^30*t^2 + 161388480*x^28*t^3 + 30502422720*x^26*t^4+  '34': x^34 + 1122*x^32*t + 556512*x^30*t^2 + 161388480*x^28*t^3 + 30502422720*x^26*t^4     + 3965314953600*x^24*t^5 + 364808975731200*x^22*t^6 + 24077392398259200*x^20*t^7     + 1143676138917312000*x^18*t^8 + 38884988723188608000*x^16*t^9 + 933239729356526592000*x^14*t^10

     + 4703528235956894023680000*x^6*t^14 + 9407056471913788047360000*x^4*t^15 + 7055292353935341035520000*x^2*t^16     + 830034394580628357120000*t^17-- params:-    m: '35'-  number: x^35 + 1190*x^33*t + 628320*x^31*t^2 + 194779200*x^29*t^3 + 39540177600*x^27*t^4+  '35': x^35 + 1190*x^33*t + 628320*x^31*t^2 + 194779200*x^29*t^3 + 39540177600*x^27*t^4     + 5551440935040*x^25*t^5 + 555144093504000*x^23*t^6 + 40128987330432000*x^21*t^7     + 2106771834847680000*x^19*t^8 + 80057329724211840000*x^17*t^9 + 2177559368498562048000*x^15*t^10

     + 23517641179784470118400000*x^7*t^14 + 65849395303396516331520000*x^5*t^15 +     82311744129245645414400000*x^3*t^16 + 29051203810321992499200000*x*t^17-- params:-    m: '36'-  number: x^36 + 1260*x^34*t + 706860*x^32*t^2 + 233735040*x^30*t^3 + 50837371200*x^28*t^4+  '36': x^36 + 1260*x^34*t + 706860*x^32*t^2 + 233735040*x^30*t^3 + 50837371200*x^28*t^4     + 7686610525440*x^26*t^5 + 832716140256000*x^24*t^6 + 65665615631616000*x^22*t^7     + 3792189302725824000*x^20*t^8 + 160114659448423680000*x^18*t^9 + 4899508579121764608000*x^16*t^10

     + 740805697163210808729600000*x^4*t^16 + 522921668585795864985600000*x^2*t^17     + 58102407620643984998400000*t^18-- params:-    m: '37'-  number: x^37 + 1332*x^35*t + 792540*x^33*t^2 + 278974080*x^31*t^3 + 64861473600*x^29*t^4+  '37': x^37 + 1332*x^35*t + 792540*x^33*t^2 + 278974080*x^31*t^3 + 64861473600*x^29*t^4     + 10533503312640*x^27*t^5 + 1232419887578880*x^25*t^6 + 105635990363904000*x^23*t^7     + 6681476390516928000*x^21*t^8 + 311802231557456640000*x^19*t^9 + 10663636319265017088000*x^17*t^10

     + 5481962159007759984599040000*x^5*t^16 + 6449367245891482334822400000*x^3*t^17     + 2149789081963827444940800000*x*t^18-- params:-    m: '38'-  number: x^38 + 1406*x^36*t + 885780*x^34*t^2 + 331281720*x^32*t^3 + 82157866560*x^30*t^4+  '38': x^38 + 1406*x^36*t + 885780*x^34*t^2 + 331281720*x^32*t^3 + 82157866560*x^30*t^4     + 14295468781440*x^28*t^5 + 1801229066461440*x^26*t^6 + 167256984742848000*x^24*t^7     + 11540731947256512000*x^22*t^8 + 592424239959167616000*x^20*t^9 + 22512121118448369408000*x^18*t^10

     + 61268988835969082180812800000*x^4*t^17 + 40845992557312721453875200000*x^2*t^18     + 4299578163927654889881600000*t^19-- params:-    m: '39'-  number: x^39 + 1482*x^37*t + 987012*x^35*t^2 + 391514760*x^33*t^3 + 103359896640*x^31*t^4+  '39': x^39 + 1482*x^37*t + 987012*x^35*t^2 + 391514760*x^33*t^3 + 103359896640*x^31*t^4     + 19224940775040*x^29*t^5 + 2601775318222080*x^27*t^6 + 260920896198842880*x^25*t^7     + 19569067214913216000*x^23*t^8 + 1100216445638454144000*x^21*t^9 + 46209090716815074048000*x^19*t^10

     + 477898112920558841010339840000*x^5*t^17 + 530997903245065378900377600000*x^3*t^18     + 167683548393178540705382400000*x*t^19-- params:-    m: '40'-  number: x^40 + 1560*x^38*t + 1096680*x^36*t^2 + 460605600*x^34*t^3 + 129199870800*x^32*t^4+  '40': x^40 + 1560*x^38*t + 1096680*x^36*t^2 + 460605600*x^34*t^3 + 129199870800*x^32*t^4     + 25633254366720*x^30*t^5 + 3716821883174400*x^28*t^6 + 401416763382835200*x^26*t^7     + 32615112024855360000*x^24*t^8 + 2000393537524462080000*x^22*t^9 + 92418181433630148096000*x^20*t^10

     + 3185987419470392273402265600000*x^6*t^17 + 5309979032450653789003776000000*x^4*t^18     + 3353670967863570814107648000000*x^2*t^19 + 335367096786357081410764800000*t^20-- params:-    m: '41'-  number: x^41 + 1640*x^39*t + 1215240*x^37*t^2 + 539566560*x^35*t^3 + 160521051600*x^33*t^4+  '41': x^41 + 1640*x^39*t + 1215240*x^37*t^2 + 539566560*x^35*t^3 + 160521051600*x^33*t^4     + 33902046097920*x^31*t^5 + 5254817145177600*x^29*t^6 + 609558788840601600*x^27*t^7     + 53488783720762790400*x^25*t^8 + 3565918914717519360000*x^23*t^9 + 180435497084706479616000*x^21*t^10

     + 18660783456898011887070412800000*x^7*t^17 + 43541828066095361069830963200000*x^5*t^18     + 45833503227468801126137856000000*x^3*t^19 + 13750050968240640337841356800000*x*t^20-- params:-    m: '42'-  number: x^42 + 1722*x^40*t + 1343160*x^38*t^2 + 629494320*x^36*t^3 + 198290710800*x^34*t^4+  '42': x^42 + 1722*x^40*t + 1343160*x^38*t^2 + 629494320*x^36*t^3 + 198290710800*x^34*t^4     + 44496435503520*x^32*t^5 + 7356744003248640*x^30*t^6 + 914338183260902400*x^28*t^7     + 86404958318155276800*x^26*t^8 + 6240358100755658880000*x^24*t^9 + 344467767161712370176000*x^22*t^10

     + 481251783888422411824447488000000*x^4*t^19 + 288751070333053447094668492800000*x^2*t^20     + 27500101936481280675682713600000*t^21-- params:-    m: '43'-  number: x^43 + 1806*x^41*t + 1480920*x^39*t^2 + 731574480*x^37*t^3 + 243614301840*x^35*t^4+  '43': x^43 + 1806*x^41*t + 1480920*x^39*t^2 + 731574480*x^37*t^3 + 243614301840*x^35*t^4     + 57980203837920*x^33*t^5 + 10204515875473920*x^31*t^6 + 1355742823455820800*x^29*t^7     + 137607896580765811200*x^27*t^8 + 10733415933299733273600*x^25*t^9 + 644004955997983996416000*x^23*t^10

     + 4138765341440432741690248396800000*x^5*t^19 + 4138765341440432741690248396800000*x^3*t^20     + 1182504383268695069054356684800000*x*t^21-- params:-    m: '44'-  number: x^44 + 1892*x^42*t + 1629012*x^40*t^2 + 847086240*x^38*t^3 + 297750813360*x^36*t^4+  '44': x^44 + 1892*x^42*t + 1629012*x^40*t^2 + 847086240*x^38*t^3 + 297750813360*x^36*t^4     + 75033204966720*x^34*t^5 + 14031209328776640*x^32*t^6 + 1988422807735203840*x^30*t^7     + 216240980341203417600*x^28*t^8 + 18164242348661087078400*x^26*t^9 + 1180675752662970660096000*x^24*t^10

     + 30350945837229840105728488243200000*x^6*t^19 + 45526418755844760158592732364800000*x^4*t^20     + 26015096431911291519195847065600000*x^2*t^21 + 2365008766537390138108713369600000*t^22-- params:-    m: '45'-  number: x^45 + 1980*x^43*t + 1787940*x^41*t^2 + 977407200*x^39*t^3 + 362129367600*x^37*t^4+  '45': x^45 + 1980*x^43*t + 1787940*x^41*t^2 + 977407200*x^39*t^3 + 362129367600*x^37*t^4     + 96471263528640*x^35*t^5 + 19133467266513600*x^33*t^6 + 2886420204776908800*x^31*t^7     + 335546348805315648000*x^29*t^8 + 30273737247768478464000*x^27*t^9 + 2125216354793347188172800*x^25*t^10

     + 195113223239334686393968852992000000*x^7*t^19 + 409737768802602841427334591283200000*x^5*t^20     + 390226446478669372787937705984000000*x^3*t^21 + 106425394494182556214892101632000000*x*t^22-- params:-    m: '46'-  number: x^46 + 2070*x^44*t + 1958220*x^42*t^2 + 1124018280*x^40*t^3 + 438367129200*x^38*t^4+  '46': x^46 + 2070*x^44*t + 1958220*x^42*t^2 + 1124018280*x^40*t^3 + 438367129200*x^38*t^4     + 123268836731040*x^36*t^5 + 25886455713518400*x^34*t^6 + 4149229044366806400*x^32*t^7     + 514504401501483993600*x^30*t^8 + 49735425478476786048000*x^28*t^9 + 3759998166172845025228800*x^26*t^10

     + 4487604134504697787061283618816000000*x^4*t^21 + 2447784073366198792942518337536000000*x^2*t^22     + 212850788988365112429784203264000000*t^23-- params:-    m: '47'-  number: x^47 + 2162*x^45*t + 2140380*x^43*t^2 + 1288508760*x^41*t^3 + 528288591600*x^39*t^4+  '47': x^47 + 2162*x^45*t + 2140380*x^43*t^2 + 1288508760*x^41*t^3 + 528288591600*x^39*t^4     + 156584738550240*x^37*t^5 + 34761811958153280*x^35*t^6 + 5909508032886057600*x^33*t^7     + 780055060340959603200*x^31*t^8 + 80605689568565825664000*x^29*t^9 + 6545181992967545043916800*x^27*t^10

     + 42183478864344159198376066016870400000*x^5*t^21 + 38348617149403781089432787288064000000*x^3*t^22     + 10003987082453160284199857553408000000*x*t^23-- params:-    m: '48'-  number: x^48 + 2256*x^46*t + 2334960*x^44*t^2 + 1472581440*x^42*t^3 + 633946309920*x^40*t^4+  '48': x^48 + 2256*x^46*t + 2334960*x^44*t^2 + 1472581440*x^42*t^3 + 633946309920*x^40*t^4     + 197791248695040*x^38*t^5 + 46349082610871040*x^36*t^6 + 8342834869956787200*x^34*t^7     + 1170082590511439404800*x^32*t^8 + 128969103309705321062400*x^30*t^9 + 11220311987944362932428800*x^28*t^10

     + 337467830914753273587008528134963200000*x^6*t^21 + 460183405792845373073193447456768000000*x^4*t^22     + 240095689978875846820796581281792000000*x^2*t^23 + 20007974164906320568399715106816000000*t^24-- params:-    m: '49'-  number: x^49 + 2352*x^47*t + 2542512*x^45*t^2 + 1678057920*x^43*t^3 + 757643150880*x^41*t^4+  '49': x^49 + 2352*x^47*t + 2542512*x^45*t^2 + 1678057920*x^43*t^3 + 757643150880*x^41*t^4     + 248506953488640*x^39*t^5 + 61381217511694080*x^37*t^6 + 11679968817939502080*x^35*t^7     + 1737395361668500934400*x^33*t^8 + 203854389102437442969600*x^31*t^9 + 18958458186526682196172800*x^29*t^10

     + 2362274816403272915109059696944742400000*x^7*t^21 + 4509797376769884656117295785076326400000*x^5*t^22     + 3921562936321638831406344160935936000000*x^3*t^23 + 980390734080409707851586040233984000000*x*t^24-- params:-    m: '50'-  number: x^50 + 2450*x^48*t + 2763600*x^46*t^2 + 1906884000*x^44*t^3 + 901956132000*x^42*t^4+  '50': x^50 + 2450*x^48*t + 2763600*x^46*t^2 + 1906884000*x^44*t^3 + 901956132000*x^42*t^4     + 310633691860800*x^40*t^5 + 80764759883808000*x^38*t^6 + 16222178913804864000*x^36*t^7     + 2554993178924266080000*x^34*t^8 + 318522482972558504640000*x^32*t^9 + 31597430310877803660288000*x^30*t^10