The generating functions for sequences enumerating (n+1,n+2) core partitions where each part can repeat at most k times for k from 1 to, 20 By Shalosh B. Ekhad The generating function enumerating (n+1,n+2)-core partitions where each par\ t can repeat at most, 1, times is 1 + t - ---------- 2 t + t - 1 and in Maple format -(1+t)/(t^2+t-1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040 , 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155, 165580141, 267914296] The generating function enumerating (n+1,n+2)-core partitions where each par\ t can repeat at most, 2, times is 2 2 t + t + 1 - ----------------- 3 2 2 t + t + t - 1 and in Maple format -(2*t^2+t+1)/(2*t^3+t^2+t-1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 9, 18, 37, 73, 146, 293, 585, 1170, 2341, 4681, 9362, 18725, 37449, 74898, 149797, 299593, 599186, 1198373, 2396745, 4793490, 9586981, 19173961, 38347922, 76695845, 153391689, 306783378, 613566757, 1227133513, 2454267026, 4908534053, 9817068105, 19634136210, 39268272421, 78536544841, 157073089682, 314146179365, 628292358729, 1256584717458] The generating function enumerating (n+1,n+2)-core partitions where each par\ t can repeat at most, 3, times is 3 2 5 t + 2 t + t + 1 - ------------------------ 4 3 2 5 t + 2 t + t + t - 1 and in Maple format -(5*t^3+2*t^2+t+1)/(5*t^4+2*t^3+t^2+t-1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 28, 62, 143, 331, 738, 1665, 3780, 8576, 19376, 43837, 99265, 224734, 508553, 1151002, 2605348, 5897126, 13347243, 30210075, 68378310, 154768501, 350303176, 792878672, 1794610400, 4061937929, 9193821553, 20809373642, 47100123053, 106606829446, 241294807548, 546148751310, 1236157833015, 2797930346651, 6332859720026, 14333849489257, 32443359067660, 73432579730224, 166207936376528] The generating function enumerating (n+1,n+2)-core partitions where each par\ t can repeat at most, 4, times is 4 3 2 14 t + 5 t + 2 t + t + 1 - -------------------------------- 5 4 3 2 14 t + 5 t + 2 t + t + t - 1 and in Maple format -(14*t^4+5*t^3+2*t^2+t+1)/(14*t^5+5*t^4+2*t^3+t^2+t-1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 90, 213, 527, 1326, 3317, 8022, 19608, 48272, 119073, 293109, 719074, 1766201, 4342666, 10679582, 26253546, 64516501, 158569355, 389788182, 958172417, 2355231458, 5789058028, 14229546200, 34976963777, 85975197161, 211329783890, 519453451997, 1276832095894, 3138498594354, 7714539273946, 18962586294533, 46610631564615, 114570438721326, 281617919565693, 692226102724158, 1701514265679040, 4182377243045824] The generating function enumerating (n+1,n+2)-core partitions where each par\ t can repeat at most, 5, times is 5 4 3 2 42 t + 14 t + 5 t + 2 t + t + 1 - ---------------------------------------- 6 5 4 3 2 42 t + 14 t + 5 t + 2 t + t + t - 1 and in Maple format -(42*t^5+14*t^4+5*t^3+2*t^2+t+1)/(42*t^6+14*t^5+5*t^4+2*t^3+t^2+t-1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 132, 297, 737, 1914, 5081, 13566, 35862, 91952, 238101, 621129, 1625770, 4254701, 11106766, 28914050, 75355116, 196604401, 513169601, 1339246650, 3493681505, 9111651086, 23767050382, 62004056984, 161765393405, 422021706953, 1100928204650, 2871929034293, 7492000533694, 19544780413050, 50987730458932, 133014063667961, 346997348696865, 905219802259962, 2361474878785257, 6160468700551902, 16071051497968006, 41925122523200704] The generating function enumerating (n+1,n+2)-core partitions where each par\ t can repeat at most, 6, times is 6 5 4 3 2 132 t + 42 t + 14 t + 5 t + 2 t + t + 1 - ------------------------------------------------- 7 6 5 4 3 2 132 t + 42 t + 14 t + 5 t + 2 t + t + t - 1 and in Maple format -(132*t^6+42*t^5+14*t^4+5*t^3+2*t^2+t+1)/(132*t^7+42*t^6+14*t^5+5*t^4+2*t^3+t^2 +t-1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 132, 429, 1001, 2574, 6929, 19110, 53286, 148580, 409437, 1097385, 2968738, 8091149, 22142494, 60664274, 166005060, 453108253, 1233972233 , 3364108566, 9181638857, 25074012710, 68477142094, 186953801156, 510221806061, 1392219793481, 3799462252970, 10370659516613, 28308833248702, 77273663928474, 210919422191140, 575679031467917, 1571229608356329, 4288536186420510, 11705450316305409, 31949972661278694, 87206889203344054, 238027540997207716] The generating function enumerating (n+1,n+2)-core partitions where each par\ t can repeat at most, 7, times is 7 6 5 4 3 2 429 t + 132 t + 42 t + 14 t + 5 t + 2 t + t + 1 - ---------------------------------------------------------- 8 7 6 5 4 3 2 429 t + 132 t + 42 t + 14 t + 5 t + 2 t + t + t - 1 and in Maple format -(429*t^7+132*t^6+42*t^5+14*t^4+5*t^3+2*t^2+t+1)/(429*t^8+132*t^7+42*t^6+14*t^5 +5*t^4+2*t^3+t^2+t-1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 132, 429, 1430, 3432, 9074, 25116, 71304, 205208, 593478, 1710855, 4870495, 13517570, 37852903, 106780058, 302575494, 859065376, 2438707193, 6911364564, 19539584312, 55140130220, 155764928680, 440512306928, 1246641740168, 3528664443392, 9986285313245, 28251589769513, 79898690830822, 225936761147253, 639001092296086, 1807541574334730, 5113448896068720, 14465811610246449, 40921535036378946, 115753870771555896, 327418092263753206, 926124136541574868] The generating function enumerating (n+1,n+2)-core partitions where each par\ t can repeat at most, 8, times is 8 7 6 5 4 3 2 1430 t + 429 t + 132 t + 42 t + 14 t + 5 t + 2 t + t + 1 - -------------------------------------------------------------------- 9 8 7 6 5 4 3 2 1430 t + 429 t + 132 t + 42 t + 14 t + 5 t + 2 t + t + t - 1 and in Maple format -(1430*t^8+429*t^7+132*t^6+42*t^5+14*t^4+5*t^3+2*t^2+t+1)/(1430*t^9+429*t^8+132 *t^7+42*t^6+14*t^5+5*t^4+2*t^3+t^2+t-1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 11934, 32266, 91324, 265268, 782238, 2324325, 6915395, 20470230, 59826283, 170804018, 491876034, 1426769656, 4158139193, 12148566534, 35516144912, 103742684220, 302464261460, 879874487328, 2555724778288, 7431048317352, 21630652147345, 63010220073053, 183606245178362, 534986867763933, 1558394569884386, 4537923907447450, 13210478734617800, 38454895040764369, 111957770800552626, 326009456617843306, 949402878243244426, 2764916238163980798] The generating function enumerating (n+1,n+2)-core partitions where each par\ t can repeat at most, 9, times is 9 8 7 6 5 4 3 2 - (4862 t + 1430 t + 429 t + 132 t + 42 t + 14 t + 5 t + 2 t + t + 1) / 10 9 8 7 6 5 4 3 / (4862 t + 1430 t + 429 t + 132 t + 42 t + 14 t + 5 t + 2 t / 2 + t + t - 1) and in Maple format -(4862*t^9+1430*t^8+429*t^7+132*t^6+42*t^5+14*t^4+5*t^3+2*t^2+t+1)/(4862*t^10+ 1430*t^9+429*t^8+132*t^7+42*t^6+14*t^5+5*t^4+2*t^3+t^2+t-1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 41990, 115634, 333336, 986442, 2966109, 9001193, 27422890, 83465327, 252466170, 754054522, 2203882564, 6495759621, 19282267966, 57515085590, 172043398988, 515237157224, 1542754462228 , 4613664432344, 13770871031720, 41020176756657, 122039589462329, 363448080371954, 1083584250136401, 3233119229973438, 9650566156166026, 28807734921595980, 85977176009107121, 256517501953483434, 765080144518648838, 2281398666763105176, 6802748383258320962] The generating function enumerating (n+1,n+2)-core partitions where each par\ t can repeat at most, 10, times is 10 9 8 7 6 5 4 3 - (16796 t + 4862 t + 1430 t + 429 t + 132 t + 42 t + 14 t + 5 t 2 / 11 10 9 8 7 + 2 t + t + 1) / (16796 t + 4862 t + 1430 t + 429 t + 132 t / 6 5 4 3 2 + 42 t + 14 t + 5 t + 2 t + t + t - 1) and in Maple format -(16796*t^10+4862*t^9+1430*t^8+429*t^7+132*t^6+42*t^5+14*t^4+5*t^3+2*t^2+t+1)/( 16796*t^11+4862*t^10+1430*t^9+429*t^8+132*t^7+42*t^6+14*t^5+5*t^4+2*t^3+t^2+t-1 ) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 149226, 417316, 1221586, 3671541, 11218265, 34628374, 107483607, 334128322, 1036160138, 3191252220, 9707423557, 28938960206, 86984272246, 263288290440, 800828245552, 2443317284244 , 7466265663824, 22822859705448, 69717024681025, 212656273294393, 647442359845482, 1967588651169297, 5973583636236814, 18153946905200106, 55230628729200092, 168165976498008097, 512266099254157626, 1560687382007232358, 4754354218049043440, 14479451346495599062] The generating function enumerating (n+1,n+2)-core partitions where each par\ t can repeat at most, 11, times is 11 10 9 8 7 6 5 4 - (58786 t + 16796 t + 4862 t + 1430 t + 429 t + 132 t + 42 t + 14 t 3 2 / 12 11 10 9 + 5 t + 2 t + t + 1) / (58786 t + 16796 t + 4862 t + 1430 t / 8 7 6 5 4 3 2 + 429 t + 132 t + 42 t + 14 t + 5 t + 2 t + t + t - 1) and in Maple format -(58786*t^11+16796*t^10+4862*t^9+1430*t^8+429*t^7+132*t^6+42*t^5+14*t^4+5*t^3+2 *t^2+t+1)/(58786*t^12+16796*t^11+4862*t^10+1430*t^9+429*t^8+132*t^7+42*t^6+14*t ^5+5*t^4+2*t^3+t^2+t-1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 534888, 1515516, 4494545, 13687277, 42388126, 132702801, 418192302, 1321977670, 4178621876, 13163217353, 41167153638, 127200597850, 385684152028, 1178933860166, 3628511588720, 11222505770352, 34821691798696, 108246887288261, 336738794812257 , 1047301848274610, 3254018635106941, 10094841644670198, 31260712963665458, 96648045168979108, 298565093031596181, 923247275307616182, 2858008379662878230, 8854506552433834124, 27446358972648586842] The generating function enumerating (n+1,n+2)-core partitions where each par\ t can repeat at most, 12, times is 12 11 10 9 8 7 6 - (208012 t + 58786 t + 16796 t + 4862 t + 1430 t + 429 t + 132 t 5 4 3 2 / 13 12 + 42 t + 14 t + 5 t + 2 t + t + 1) / (208012 t + 58786 t / 11 10 9 8 7 6 5 4 + 16796 t + 4862 t + 1430 t + 429 t + 132 t + 42 t + 14 t + 5 t 3 2 + 2 t + t + t - 1) and in Maple format -(208012*t^12+58786*t^11+16796*t^10+4862*t^9+1430*t^8+429*t^7+132*t^6+42*t^5+14 *t^4+5*t^3+2*t^2+t+1)/(208012*t^13+58786*t^12+16796*t^11+4862*t^10+1430*t^9+429 *t^8+132*t^7+42*t^6+14*t^5+5*t^4+2*t^3+t^2+t-1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 1931540, 5534605, 16599445, 51124630, 160160385, 507429450, 1619434830, 5189976220, 16656986905, 53395347070, 170469589994, 540216266828, 1691980481302, 5206284480828, 16148080577736, 50425072133784, 158230275898941, 498166544022369 , 1571642499255666, 4963336497158013, 15676658659213422, 49485255289793778, 156023405310608052, 491163490958670453, 1543575973974977670, 4843984203400945510, 15191496456596376700, 47690201812121061050] The generating function enumerating (n+1,n+2)-core partitions where each par\ t can repeat at most, 13, times is 13 12 11 10 9 8 7 - (742900 t + 208012 t + 58786 t + 16796 t + 4862 t + 1430 t + 429 t 6 5 4 3 2 / 14 + 132 t + 42 t + 14 t + 5 t + 2 t + t + 1) / (742900 t / 13 12 11 10 9 8 + 208012 t + 58786 t + 16796 t + 4862 t + 1430 t + 429 t 7 6 5 4 3 2 + 132 t + 42 t + 14 t + 5 t + 2 t + t + t - 1) and in Maple format -(742900*t^13+208012*t^12+58786*t^11+16796*t^10+4862*t^9+1430*t^8+429*t^7+132*t ^6+42*t^5+14*t^4+5*t^3+2*t^2+t+1)/(742900*t^14+208012*t^13+58786*t^12+16796*t^ 11+4862*t^10+1430*t^9+429*t^8+132*t^7+42*t^6+14*t^5+5*t^4+2*t^3+t^2+t-1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 7020405, 20313945, 61525230, 191362185, 605492250, 1938138930, 6252323220, 20268966705, 65873095470, 214141709394, 694748381628, 2243880891302, 7193125956828, 22798480518236, 71062900994784, 223250637142941, 706039748015369 , 2243737250001666, 7154582747918013, 22864171837799422, 73158624521162778, 234186239468744052, 749452856186390053, 2396442659546902470, 7653133709023537110, 24403018470731019100, 77689619768250059850] The generating function enumerating (n+1,n+2)-core partitions where each par\ t can repeat at most, 14, times is 14 13 12 11 10 9 - (2674440 t + 742900 t + 208012 t + 58786 t + 16796 t + 4862 t 8 7 6 5 4 3 2 / + 1430 t + 429 t + 132 t + 42 t + 14 t + 5 t + 2 t + t + 1) / ( / 15 14 13 12 11 10 2674440 t + 742900 t + 208012 t + 58786 t + 16796 t + 4862 t 9 8 7 6 5 4 3 2 + 1430 t + 429 t + 132 t + 42 t + 14 t + 5 t + 2 t + t + t - 1) and in Maple format -(2674440*t^14+742900*t^13+208012*t^12+58786*t^11+16796*t^10+4862*t^9+1430*t^8+ 429*t^7+132*t^6+42*t^5+14*t^4+5*t^3+2*t^2+t+1)/(2674440*t^15+742900*t^14+208012 *t^13+58786*t^12+16796*t^11+4862*t^10+1430*t^9+429*t^8+132*t^7+42*t^6+14*t^5+5* t^4+2*t^3+t^2+t-1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 25662825, 74897430, 228804345, 717818730, 2291165010, 7399657980, 24093415905, 78876222750, 259061603634, 851968011468, 2800196504582, 9179967432828, 29951109831836, 96991182256584, 310659974784141, 979452509718569 , 3112086009526866, 9953213204224413, 31986486897131422, 103146831187687578, 333388990590218052, 1079097849134538853, 3495071326172435430, 11320281088058106390, 36646052616338310460, 118514618391980689290] The generating function enumerating (n+1,n+2)-core partitions where each par\ t can repeat at most, 15, times is 15 14 13 12 11 10 - (9694845 t + 2674440 t + 742900 t + 208012 t + 58786 t + 16796 t 9 8 7 6 5 4 3 2 + 4862 t + 1430 t + 429 t + 132 t + 42 t + 14 t + 5 t + 2 t + t / 16 15 14 13 12 + 1) / (9694845 t + 2674440 t + 742900 t + 208012 t + 58786 t / 11 10 9 8 7 6 5 4 + 16796 t + 4862 t + 1430 t + 429 t + 132 t + 42 t + 14 t + 5 t 3 2 + 2 t + t + t - 1) and in Maple format -(9694845*t^15+2674440*t^14+742900*t^13+208012*t^12+58786*t^11+16796*t^10+4862* t^9+1430*t^8+429*t^7+132*t^6+42*t^5+14*t^4+5*t^3+2*t^2+t+1)/(9694845*t^16+ 2674440*t^15+742900*t^14+208012*t^13+58786*t^12+16796*t^11+4862*t^10+1430*t^9+ 429*t^8+132*t^7+42*t^6+14*t^5+5*t^4+2*t^3+t^2+t-1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 94287120, 277278570, 853546560, 2698348500, 8679377520, 28252504410, 92739851100, 306197940024, 1014802628088, 3370117662752, 11196611530968, 37153410182336, 122919463518384, 404649994358166, 1322239639929719, 4274982134060391, 13616302047581538, 43703424352738747, 141180489139400778, 458256041052787302, 1492577067306429328, 4873104296493571455, 15934811860332038370, 52149867289118020450, 170711431791293797020] The generating function enumerating (n+1,n+2)-core partitions where each par\ t can repeat at most, 16, times is 16 15 14 13 12 - (35357670 t + 9694845 t + 2674440 t + 742900 t + 208012 t 11 10 9 8 7 6 5 + 58786 t + 16796 t + 4862 t + 1430 t + 429 t + 132 t + 42 t 4 3 2 / 17 16 + 14 t + 5 t + 2 t + t + 1) / (35357670 t + 9694845 t / 15 14 13 12 11 10 + 2674440 t + 742900 t + 208012 t + 58786 t + 16796 t + 4862 t 9 8 7 6 5 4 3 2 + 1430 t + 429 t + 132 t + 42 t + 14 t + 5 t + 2 t + t + t - 1) and in Maple format -(35357670*t^16+9694845*t^15+2674440*t^14+742900*t^13+208012*t^12+58786*t^11+ 16796*t^10+4862*t^9+1430*t^8+429*t^7+132*t^6+42*t^5+14*t^4+5*t^3+2*t^2+t+1)/( 35357670*t^17+9694845*t^16+2674440*t^15+742900*t^14+208012*t^13+58786*t^12+ 16796*t^11+4862*t^10+1430*t^9+429*t^8+132*t^7+42*t^6+14*t^5+5*t^4+2*t^3+t^2+t-1 ) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 347993910, 1030334910, 3193355880, 10164399660, 32919716850, 107908291530, 356759408124, 1186711619628, 3963985088072, 13275147519588, 44508229834376, 149186676561384, 499211961312966, 1665026770140869, 5525146961889291, 18200239749620838, 59341451058738847, 190748427927002778, 617816551973054802, 2013830002362247528, 6595409939650719855, 21675008035062725970, 71407192882129050430, 235636143193068447060] The generating function enumerating (n+1,n+2)-core partitions where each par\ t can repeat at most, 17, times is 17 16 15 14 13 - (129644790 t + 35357670 t + 9694845 t + 2674440 t + 742900 t 12 11 10 9 8 7 6 + 208012 t + 58786 t + 16796 t + 4862 t + 1430 t + 429 t + 132 t 5 4 3 2 / 18 17 + 42 t + 14 t + 5 t + 2 t + t + 1) / (129644790 t + 35357670 t / 16 15 14 13 12 + 9694845 t + 2674440 t + 742900 t + 208012 t + 58786 t 11 10 9 8 7 6 5 4 + 16796 t + 4862 t + 1430 t + 429 t + 132 t + 42 t + 14 t + 5 t 3 2 + 2 t + t + t - 1) and in Maple format -(129644790*t^17+35357670*t^16+9694845*t^15+2674440*t^14+742900*t^13+208012*t^ 12+58786*t^11+16796*t^10+4862*t^9+1430*t^8+429*t^7+132*t^6+42*t^5+14*t^4+5*t^3+ 2*t^2+t+1)/(129644790*t^18+35357670*t^17+9694845*t^16+2674440*t^15+742900*t^14+ 208012*t^13+58786*t^12+16796*t^11+4862*t^10+1430*t^9+429*t^8+132*t^7+42*t^6+14* t^5+5*t^4+2*t^3+t^2+t-1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1289624490, 3841579830, 11979426720, 38364798030, 125021403810, 412377023034, 1372103669328, 4594318057052, 15452661412428, 52129528459316, 176154348618864, 595525075803966, 2011753982308469, 6782033105996841, 22784177451660138, 76149222632882947, 252671796884375778, 830125245541190802, 2690563963110681028, 8786290901346450555, 28873300723898508570, 95328707795320212730, 315826766831371027320] The generating function enumerating (n+1,n+2)-core partitions where each par\ t can repeat at most, 18, times is 18 17 16 15 14 - (477638700 t + 129644790 t + 35357670 t + 9694845 t + 2674440 t 13 12 11 10 9 8 + 742900 t + 208012 t + 58786 t + 16796 t + 4862 t + 1430 t 7 6 5 4 3 2 / + 429 t + 132 t + 42 t + 14 t + 5 t + 2 t + t + 1) / ( / 19 18 17 16 15 477638700 t + 129644790 t + 35357670 t + 9694845 t + 2674440 t 14 13 12 11 10 9 + 742900 t + 208012 t + 58786 t + 16796 t + 4862 t + 1430 t 8 7 6 5 4 3 2 + 429 t + 132 t + 42 t + 14 t + 5 t + 2 t + t + t - 1) and in Maple format -(477638700*t^18+129644790*t^17+35357670*t^16+9694845*t^15+2674440*t^14+742900* t^13+208012*t^12+58786*t^11+16796*t^10+4862*t^9+1430*t^8+429*t^7+132*t^6+42*t^5 +14*t^4+5*t^3+2*t^2+t+1)/(477638700*t^19+129644790*t^18+35357670*t^17+9694845*t ^16+2674440*t^15+742900*t^14+208012*t^13+58786*t^12+16796*t^11+4862*t^10+1430*t ^9+429*t^8+132*t^7+42*t^6+14*t^5+5*t^4+2*t^3+t^2+t-1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 4796857230, 14367620220, 45051739830, 145082229210, 475425331434, 1577010671628, 5277341398052, 17774940771828, 60151948064516, 204232817237064, 694879657068366, 2366591772538469, 8059449150824841, 27414810614161638, 93037414166711947, 314595165841748778, 1058263973278880802, 3534677255740134028, 11693430117661014555, 38186693928712206570, 125635836371476644730, 415919651462788936320] The generating function enumerating (n+1,n+2)-core partitions where each par\ t can repeat at most, 19, times is 19 18 17 16 15 - (1767263190 t + 477638700 t + 129644790 t + 35357670 t + 9694845 t 14 13 12 11 10 9 + 2674440 t + 742900 t + 208012 t + 58786 t + 16796 t + 4862 t 8 7 6 5 4 3 2 / + 1430 t + 429 t + 132 t + 42 t + 14 t + 5 t + 2 t + t + 1) / ( / 20 19 18 17 16 1767263190 t + 477638700 t + 129644790 t + 35357670 t + 9694845 t 15 14 13 12 11 10 + 2674440 t + 742900 t + 208012 t + 58786 t + 16796 t + 4862 t 9 8 7 6 5 4 3 2 + 1430 t + 429 t + 132 t + 42 t + 14 t + 5 t + 2 t + t + t - 1) and in Maple format -(1767263190*t^19+477638700*t^18+129644790*t^17+35357670*t^16+9694845*t^15+ 2674440*t^14+742900*t^13+208012*t^12+58786*t^11+16796*t^10+4862*t^9+1430*t^8+ 429*t^7+132*t^6+42*t^5+14*t^4+5*t^3+2*t^2+t+1)/(1767263190*t^20+477638700*t^19+ 129644790*t^18+35357670*t^17+9694845*t^16+2674440*t^15+742900*t^14+208012*t^13+ 58786*t^12+16796*t^11+4862*t^10+1430*t^9+429*t^8+132*t^7+42*t^6+14*t^5+5*t^4+2* t^3+t^2+t-1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 17902146600, 53888055780, 169823913870, 549650385414, 1810289412708, 6035497306562, 20302127133528, 68744381694296, 233915769776304, 798769990955706, 2734203723216749, 9372348974675841, 32141249980025238, 110170756867967497, 377081474516916078, 1287380438421160902, 4378790548369587028, 14816649300389990655, 49787222321705546370, 165750950289904662430, 545022704376419740020] The generating function enumerating (n+1,n+2)-core partitions where each par\ t can repeat at most, 20, times is 20 19 18 17 - (6564120420 t + 1767263190 t + 477638700 t + 129644790 t 16 15 14 13 12 + 35357670 t + 9694845 t + 2674440 t + 742900 t + 208012 t 11 10 9 8 7 6 5 + 58786 t + 16796 t + 4862 t + 1430 t + 429 t + 132 t + 42 t 4 3 2 / 21 20 + 14 t + 5 t + 2 t + t + 1) / (6564120420 t + 1767263190 t / 19 18 17 16 15 + 477638700 t + 129644790 t + 35357670 t + 9694845 t + 2674440 t 14 13 12 11 10 9 + 742900 t + 208012 t + 58786 t + 16796 t + 4862 t + 1430 t 8 7 6 5 4 3 2 + 429 t + 132 t + 42 t + 14 t + 5 t + 2 t + t + t - 1) and in Maple format -(6564120420*t^20+1767263190*t^19+477638700*t^18+129644790*t^17+35357670*t^16+ 9694845*t^15+2674440*t^14+742900*t^13+208012*t^12+58786*t^11+16796*t^10+4862*t^ 9+1430*t^8+429*t^7+132*t^6+42*t^5+14*t^4+5*t^3+2*t^2+t+1)/(6564120420*t^21+ 1767263190*t^20+477638700*t^19+129644790*t^18+35357670*t^17+9694845*t^16+ 2674440*t^15+742900*t^14+208012*t^13+58786*t^12+16796*t^11+4862*t^10+1430*t^9+ 429*t^8+132*t^7+42*t^6+14*t^5+5*t^4+2*t^3+t^2+t-1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 67016296620, 202644515970, 641548071294, 2085982470348, 6901961202002, 23118134793708, 78131073894896, 265830523258344, 909020957530026, 3120082106226869, 10737764791480881, 37017735040043238, 127726103084032297, 440719604550150978, 1519472442071782302, 5229794561755198828, 17951927244442244655, 61387750714698886170, 208838627178165638830, 705622227323574288420] This ends this article, that took, 0.122, seconds to generate.