Heat (caloric) polynomials $P_m(x,t)$
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Polynomials
$m$ 
$P_m(x,t)$
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 + 15440875522080712704000*x^12*t^11 + 169849630742887839744000*x^10*t^12 + 1175882058989223505920000*x^8*t^13 + 4703528235956894023680000*x^6*t^14 + 9407056471913788047360000*x^4*t^15 + 7055292353935341035520000*x^2*t^16 + 830034394580628357120000*t^17
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 + 41571587944063457280000*x^13*t^11 + 540430643272824944640000*x^11*t^12 + 4572874673846980300800000*x^9*t^13 + 23517641179784470118400000*x^7*t^14 + 65849395303396516331520000*x^5*t^15 + 82311744129245645414400000*x^3*t^16 + 29051203810321992499200000*x*t^17
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 + 106898368999020318720000*x^14*t^11 + 1621291929818474833920000*x^12*t^12 + 16462348825849129082880000*x^10*t^13 + 105829385309030115532800000*x^8*t^14 + 395096371820379097989120000*x^6*t^15 + 740805697163210808729600000*x^4*t^16 + 522921668585795864985600000*x^2*t^17 + 58102407620643984998400000*t^18
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 + 263682643530916786176000*x^15*t^11 + 4614446261791043758080000*x^13*t^12 + 55373355141492525096960000*x^11*t^13 + 435076361826012697190400000*x^9*t^14 + 2088366536764860946513920000*x^7*t^15 + 5481962159007759984599040000*x^5*t^16 + 6449367245891482334822400000*x^3*t^17 + 2149789081963827444940800000*x*t^18
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 + 626246278385927367168000*x^16*t^11 + 12524925567718547343360000*x^14*t^12 + 175348957948059662807040000*x^12*t^13 + 1653290174938848249323520000*x^10*t^14 + 9919741049633089495941120000*x^8*t^15 + 34719093673715813235793920000*x^6*t^16 + 61268988835969082180812800000*x^4*t^17 + 40845992557312721453875200000*x^2*t^18 + 4299578163927654889881600000*t^19
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 + 1436682638650068665856000*x^17*t^11 + 32564806476068223092736000*x^15*t^12 + 526046873844178988421120000*x^13*t^13 + 5861665165692280156692480000*x^11*t^14 + 42985544548410054482411520000*x^9*t^15 + 193434950467845245170851840000*x^7*t^16 + 477898112920558841010339840000*x^5*t^17 + 530997903245065378900377600000*x^3*t^18 + 167683548393178540705382400000*x*t^19
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 + 3192628085889041479680000*x^18*t^11 + 81412016190170557731840000*x^16*t^12 + 1502991068126225681203200000*x^14*t^13 + 19538883885640933855641600000*x^12*t^14 + 171942178193640217929646080000*x^10*t^15 + 967174752339226225854259200000*x^8*t^16 + 3185987419470392273402265600000*x^6*t^17 + 5309979032450653789003776000000*x^4*t^18 + 3353670967863570814107648000000*x^2*t^19 + 335367096786357081410764800000*t^20
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 + 6889355343234247403520000*x^19*t^11 + 196346627282176051000320000*x^17*t^12 + 4108175586211683528622080000*x^15*t^13 + 61622633793175252929331200000*x^13*t^14 + 640875391449022630465044480000*x^11*t^15 + 4406018316212030584447180800000*x^9*t^16 + 18660783456898011887070412800000*x^7*t^17 + 43541828066095361069830963200000*x^5*t^18 + 45833503227468801126137856000000*x^3*t^19 + 13750050968240640337841356800000*x*t^20
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 + 14467646220791919547392000*x^20*t^11 + 458142130325077452334080000*x^18*t^12 + 10783960913805669262632960000*x^16*t^13 + 184867901379525758787993600000*x^14*t^14 + 2243063870071579206627655680000*x^12*t^15 + 18505276928090528454678159360000*x^10*t^16 + 97969113148714562407119667200000*x^8*t^17 + 304792796462667527488816742400000*x^6*t^18 + 481251783888422411824447488000000*x^4*t^19 + 288751070333053447094668492800000*x^2*t^20 + 27500101936481280675682713600000*t^21
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 + 29624227975907263835136000*x^21*t^11 + 1036847979156754234229760000*x^19*t^12 + 27277077605508457546659840000*x^17*t^13 + 529954650621307175192248320000*x^15*t^14 + 7419365108698300452691476480000*x^13*t^15 + 72338809809808429413741895680000*x^11*t^16 + 468074651710525131500682854400000*x^9*t^17 + 1872298606842100526002731417600000*x^7*t^18 + 4138765341440432741690248396800000*x^5*t^19 + 4138765341440432741690248396800000*x^3*t^20 + 1182504383268695069054356684800000*x*t^21
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 + 59248455951814527670272000*x^22*t^11 + 2281065554144859315305472000*x^20*t^12 + 66677300813465118447390720000*x^18*t^13 + 1457375289208594731778682880000*x^16*t^14 + 23318004627337515708458926080000*x^14*t^15 + 265242302635964241183720284160000*x^12*t^16 + 2059528467526310578603004559360000*x^10*t^17 + 10297642337631552893015022796800000*x^8*t^18 + 30350945837229840105728488243200000*x^6*t^19 + 45526418755844760158592732364800000*x^4*t^20 + 26015096431911291519195847065600000*x^2*t^21 + 2365008766537390138108713369600000*t^22
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 + 115920892079637119354880000*x^23*t^11 + 4887997616024698532797440000*x^21*t^12 + 157919922979259491059609600000*x^19*t^13 + 3857758118493338995884748800000*x^17*t^14 + 69954013882012547125376778240000*x^15*t^15 + 918146432201414681020570214400000*x^13*t^16 + 8425343730789452367012291379200000*x^11*t^17 + 51488211688157764465075113984000000*x^9*t^18 + 195113223239334686393968852992000000*x^7*t^19 + 409737768802602841427334591283200000*x^5*t^20 + 390226446478669372787937705984000000*x^3*t^21 + 106425394494182556214892101632000000*x*t^22
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 + 222181709819304478763520000*x^24*t^11 + 10220358651688006023121920000*x^22*t^12 + 363215822852296829437102080000*x^20*t^13 + 9858715191705199656149913600000*x^18*t^14 + 201117789910786072985458237440000*x^16*t^15 + 3016766848661791094781873561600000*x^14*t^16 + 32297150968026234073547116953600000*x^12*t^17 + 236845773765525716539345524326400000*x^10*t^18 + 1121901033626174446765320904704000000*x^8*t^19 + 3141322894153288450942898533171200000*x^6*t^20 + 4487604134504697787061283618816000000*x^4*t^21 + 2447784073366198792942518337536000000*x^2*t^22 + 212850788988365112429784203264000000*t^23
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 + 417701614460292420075417600*x^25*t^11 + 20885080723014621003770880000*x^23*t^12 + 812911603526569094454466560000*x^21*t^13 + 24387348105797072833633996800000*x^19*t^14 + 556031536812173260606855127040000*x^17*t^15 + 9452536125806945430316537159680000*x^15*t^16 + 116766622730556384727439576678400000*x^13*t^17 + 1011977396998155334304476331212800000*x^11*t^18 + 5858816508936688777552231391232000000*x^9*t^19 + 21091739432172079599188033008435200000*x^7*t^20 + 42183478864344159198376066016870400000*x^5*t^21 + 38348617149403781089432787288064000000*x^3*t^22 + 10003987082453160284199857553408000000*x*t^23
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 + 771141442080539852446924800*x^26*t^11 + 41770161446029242007541760000*x^24*t^12 + 1773625316785241660627927040000*x^22*t^13 + 58529635453912974800721592320000*x^20*t^14 + 1482750764832462028284947005440000*x^18*t^15 + 28357608377420836290949611479040000*x^16*t^16 + 400342706504764747636935691468800000*x^14*t^17 + 4047909587992621337217905324851200000*x^12*t^18 + 28122319242896106132250710677913600000*x^10*t^19 + 126550436593032477595128198050611200000*x^8*t^20 + 337467830914753273587008528134963200000*x^6*t^21 + 460183405792845373073193447456768000000*x^4*t^22 + 240095689978875846820796581281792000000*x^2*t^23 + 20007974164906320568399715106816000000*t^24
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 + 1399478913405424176662937600*x^27*t^11 + 81869516434217314334781849600*x^25*t^12 + 3778593066194645276989931520000*x^23*t^13 + 136569149392463607868350382080000*x^21*t^14 + 3823936182988981020313810698240000*x^19*t^15 + 81736635911389469309207703674880000*x^17*t^16 + 1307786174582231508947323258798080000*x^15*t^17 + 15257505370126034271052104685977600000*x^13*t^18 + 125272149354719018225480438474342400000*x^11*t^19 + 688996821450954600240142411608883200000*x^9*t^20 + 2362274816403272915109059696944742400000*x^7*t^21 + 4509797376769884656117295785076326400000*x^5*t^22 + 3921562936321638831406344160935936000000*x^3*t^23 + 980390734080409707851586040233984000000*x*t^24
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 + 2499069488223971744040960000*x^28*t^11 + 157441377758110219874580480000*x^26*t^12 + 7872068887905510993729024000000*x^24*t^13 + 310384430437417290609887232000000*x^22*t^14 + 9559840457472452550784526745600000*x^20*t^15 + 227046210864970748081132510208000000*x^18*t^16 + 4086831795569473465460385183744000000*x^16*t^17 + 54491090607592979539471802449920000000*x^14*t^18 + 521967288977995909272835160309760000000*x^12*t^19 + 3444984107254773001200712058044416000000*x^10*t^20 + 14764217602520455719431623105904640000000*x^8*t^21 + 37581644806415705467644131542302720000000*x^6*t^22 + 49019536704020485392579302011699200000000*x^4*t^23 + 24509768352010242696289651005849600000000*x^2*t^24 + 1960781468160819415703172080467968000000*t^25
Definition
The heat polynomial $P_m(x,t)$ is the unique polynomial solution of $\partial P/\partial t=\partial^2P/\partial x^2$ with $P_m(x,0)=x^m$ [1].
Parameters
$m$
—   degree ($m\geq0$)
Formulas
(1)
$P_m(x,t)=\sum_{\ell=0}^{\lfloor m/2\rfloor} \frac{m!}{\ell!(m-2\ell)!}x^{m-2\ell}t^\ell$.
(2)
$P_m(\lambda x,\lambda^2t)=\lambda^mP_m(x,t)$ for $\lambda>0$; this is parabolic $m$-homogeneity.
(3)
$P_0(x,t)=1$ and $P_{m+1}(x,t)=xP_m(x,t)+2t\,\partial P_m(x,t)/\partial x$.
(4)
$P_m(x,-1/2)=He_m(x)$, where $He_m$ is the probabilists' Hermite polynomial.
Programs
(P1)
Sage
from sage.all import ZZ, PolynomialRing

R = PolynomialRing(ZZ, ('x', 't'))
x, t = R.gens()

def heat_polynomial(m):
    p = R.one()
    for _ in range(m):
        p = x * p + ZZ(2) * t * p.derivative(x)
    return p

heat_polynomial(51)      # the next one after this table
References
[1]
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) (zbMATH)
Links
Similar tables
Hermite polynomials in probabilist's convention —   are the specialization $P_m(x,-1/2)$ in (4)
Data properties
Entries are of type: integral polynomial
Table is complete: no (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)
How they were obtained:

The generator computes (3) over Sage's integer polynomial ring.

more

The values were checked against (1), against the heat equation, against the normalization $P_m(x,0)=x^m$, against the parabolic homogeneity identity (2), and against the specialization to probabilists' Hermite polynomials in (4).