240225

PARI


Expansion of (1/x) * Series_Reversion( x * (1-x)^u * (1-x^s)^t ).

2通りで算出し、一致することを確認してみた。

a(n, s, t, u) = sum(k=0, n\s, binomial(t*(n+1)+k-1, k)*binomial((u+1)*(n+1)-s*k-2, n-s*k))/(n+1);
b(n, s, t, u) = my(x='x+O('x^(n+2))); polcoef(serreverse(x*(1-x)^u*(1-x^s)^t)/x, n);
diff(n, s, t, u) = a(n, s, t, u)-b(n, s, t, u);

for(s=1, 2, for(t=-5, 5, for(u=-5, 5, print1([s, t, u],": "); for(n=0, 10, if(diff(n, s, t, u)==0, print1(a(n, s, t, u),", "))); print)))

出力結果

[1, -5, -5]: 1, -10, 145, -2470, 46060, -910252, 18730855, -397089550, 8612835715, -190223180840, 4263421511271, 

[1, -5, -4]: 1, -9, 117, -1785, 29799, -527085, 9706503, -184138713, 3573805950, -70625252863, 1416298046436, 

[1, -5, -3]: 1, -8, 92, -1240, 18278, -285384, 4638348, -77652024, 1329890705, -23190029720, 410333440536, 

[1, -5, -2]: 1, -7, 70, -819, 10472, -141778, 1997688, -28989675, 430321633, -6503352856, 99726673130, 

[1, -5, -1]: 1, -6, 51, -506, 5481, -62832, 749398, -9203634, 115607310, -1478314266, 19180049928, 

[1, -5, 0]: 1, -5, 35, -285, 2530, -23751, 231880, -2330445, 23950355, -250543370, 2658968130, 

[1, -5, 1]: 1, -4, 22, -140, 969, -7084, 53820, -420732, 3362260, -27343888, 225568798, 

[1, -5, 2]: 1, -3, 12, -55, 273, -1428, 7752, -43263, 246675, -1430715, 8414640, 

[1, -5, 3]: 1, -2, 5, -14, 42, -132, 429, -1430, 4862, -16796, 58786, 

[1, -5, 4]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 

[1, -5, 5]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

[1, -4, -5]: 1, -9, 117, -1785, 29799, -527085, 9706503, -184138713, 3573805950, -70625252863, 1416298046436, 

[1, -4, -4]: 1, -8, 92, -1240, 18278, -285384, 4638348, -77652024, 1329890705, -23190029720, 410333440536, 

[1, -4, -3]: 1, -7, 70, -819, 10472, -141778, 1997688, -28989675, 430321633, -6503352856, 99726673130, 

[1, -4, -2]: 1, -6, 51, -506, 5481, -62832, 749398, -9203634, 115607310, -1478314266, 19180049928, 

[1, -4, -1]: 1, -5, 35, -285, 2530, -23751, 231880, -2330445, 23950355, -250543370, 2658968130, 

[1, -4, 0]: 1, -4, 22, -140, 969, -7084, 53820, -420732, 3362260, -27343888, 225568798, 

[1, -4, 1]: 1, -3, 12, -55, 273, -1428, 7752, -43263, 246675, -1430715, 8414640, 

[1, -4, 2]: 1, -2, 5, -14, 42, -132, 429, -1430, 4862, -16796, 58786, 

[1, -4, 3]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 

[1, -4, 4]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

[1, -4, 5]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[1, -3, -5]: 1, -8, 92, -1240, 18278, -285384, 4638348, -77652024, 1329890705, -23190029720, 410333440536, 

[1, -3, -4]: 1, -7, 70, -819, 10472, -141778, 1997688, -28989675, 430321633, -6503352856, 99726673130, 

[1, -3, -3]: 1, -6, 51, -506, 5481, -62832, 749398, -9203634, 115607310, -1478314266, 19180049928, 

[1, -3, -2]: 1, -5, 35, -285, 2530, -23751, 231880, -2330445, 23950355, -250543370, 2658968130, 

[1, -3, -1]: 1, -4, 22, -140, 969, -7084, 53820, -420732, 3362260, -27343888, 225568798, 

[1, -3, 0]: 1, -3, 12, -55, 273, -1428, 7752, -43263, 246675, -1430715, 8414640, 

[1, -3, 1]: 1, -2, 5, -14, 42, -132, 429, -1430, 4862, -16796, 58786, 

[1, -3, 2]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 

[1, -3, 3]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

[1, -3, 4]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[1, -3, 5]: 1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 

[1, -2, -5]: 1, -7, 70, -819, 10472, -141778, 1997688, -28989675, 430321633, -6503352856, 99726673130, 

[1, -2, -4]: 1, -6, 51, -506, 5481, -62832, 749398, -9203634, 115607310, -1478314266, 19180049928, 

[1, -2, -3]: 1, -5, 35, -285, 2530, -23751, 231880, -2330445, 23950355, -250543370, 2658968130, 

[1, -2, -2]: 1, -4, 22, -140, 969, -7084, 53820, -420732, 3362260, -27343888, 225568798, 

[1, -2, -1]: 1, -3, 12, -55, 273, -1428, 7752, -43263, 246675, -1430715, 8414640, 

[1, -2, 0]: 1, -2, 5, -14, 42, -132, 429, -1430, 4862, -16796, 58786, 

[1, -2, 1]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 

[1, -2, 2]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

[1, -2, 3]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[1, -2, 4]: 1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 

[1, -2, 5]: 1, 3, 15, 91, 612, 4389, 32890, 254475, 2017356, 16301164, 133767543, 

[1, -1, -5]: 1, -6, 51, -506, 5481, -62832, 749398, -9203634, 115607310, -1478314266, 19180049928, 

[1, -1, -4]: 1, -5, 35, -285, 2530, -23751, 231880, -2330445, 23950355, -250543370, 2658968130, 

[1, -1, -3]: 1, -4, 22, -140, 969, -7084, 53820, -420732, 3362260, -27343888, 225568798, 

[1, -1, -2]: 1, -3, 12, -55, 273, -1428, 7752, -43263, 246675, -1430715, 8414640, 

[1, -1, -1]: 1, -2, 5, -14, 42, -132, 429, -1430, 4862, -16796, 58786, 

[1, -1, 0]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 

[1, -1, 1]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

[1, -1, 2]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[1, -1, 3]: 1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 

[1, -1, 4]: 1, 3, 15, 91, 612, 4389, 32890, 254475, 2017356, 16301164, 133767543, 

[1, -1, 5]: 1, 4, 26, 204, 1771, 16380, 158224, 1577532, 16112057, 167710664, 1772645420, 

[1, 0, -5]: 1, -5, 35, -285, 2530, -23751, 231880, -2330445, 23950355, -250543370, 2658968130, 

[1, 0, -4]: 1, -4, 22, -140, 969, -7084, 53820, -420732, 3362260, -27343888, 225568798, 

[1, 0, -3]: 1, -3, 12, -55, 273, -1428, 7752, -43263, 246675, -1430715, 8414640, 

[1, 0, -2]: 1, -2, 5, -14, 42, -132, 429, -1430, 4862, -16796, 58786, 

[1, 0, -1]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 

[1, 0, 0]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

[1, 0, 1]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[1, 0, 2]: 1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 

[1, 0, 3]: 1, 3, 15, 91, 612, 4389, 32890, 254475, 2017356, 16301164, 133767543, 

[1, 0, 4]: 1, 4, 26, 204, 1771, 16380, 158224, 1577532, 16112057, 167710664, 1772645420, 

[1, 0, 5]: 1, 5, 40, 385, 4095, 46376, 548340, 6690585, 83615350, 1064887395, 13770292256, 

[1, 1, -5]: 1, -4, 22, -140, 969, -7084, 53820, -420732, 3362260, -27343888, 225568798, 

[1, 1, -4]: 1, -3, 12, -55, 273, -1428, 7752, -43263, 246675, -1430715, 8414640, 

[1, 1, -3]: 1, -2, 5, -14, 42, -132, 429, -1430, 4862, -16796, 58786, 

[1, 1, -2]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 

[1, 1, -1]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

[1, 1, 0]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[1, 1, 1]: 1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 

[1, 1, 2]: 1, 3, 15, 91, 612, 4389, 32890, 254475, 2017356, 16301164, 133767543, 

[1, 1, 3]: 1, 4, 26, 204, 1771, 16380, 158224, 1577532, 16112057, 167710664, 1772645420, 

[1, 1, 4]: 1, 5, 40, 385, 4095, 46376, 548340, 6690585, 83615350, 1064887395, 13770292256, 

[1, 1, 5]: 1, 6, 57, 650, 8184, 109668, 1533939, 22137570, 327203085, 4928006512, 75357373305, 

[1, 2, -5]: 1, -3, 12, -55, 273, -1428, 7752, -43263, 246675, -1430715, 8414640, 

[1, 2, -4]: 1, -2, 5, -14, 42, -132, 429, -1430, 4862, -16796, 58786, 

[1, 2, -3]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 

[1, 2, -2]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

[1, 2, -1]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[1, 2, 0]: 1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 

[1, 2, 1]: 1, 3, 15, 91, 612, 4389, 32890, 254475, 2017356, 16301164, 133767543, 

[1, 2, 2]: 1, 4, 26, 204, 1771, 16380, 158224, 1577532, 16112057, 167710664, 1772645420, 

[1, 2, 3]: 1, 5, 40, 385, 4095, 46376, 548340, 6690585, 83615350, 1064887395, 13770292256, 

[1, 2, 4]: 1, 6, 57, 650, 8184, 109668, 1533939, 22137570, 327203085, 4928006512, 75357373305, 

[1, 2, 5]: 1, 7, 77, 1015, 14763, 228459, 3689595, 61474519, 1048927880, 18236463245, 321899509386, 

[1, 3, -5]: 1, -2, 5, -14, 42, -132, 429, -1430, 4862, -16796, 58786, 

[1, 3, -4]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 

[1, 3, -3]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

[1, 3, -2]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[1, 3, -1]: 1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 

[1, 3, 0]: 1, 3, 15, 91, 612, 4389, 32890, 254475, 2017356, 16301164, 133767543, 

[1, 3, 1]: 1, 4, 26, 204, 1771, 16380, 158224, 1577532, 16112057, 167710664, 1772645420, 

[1, 3, 2]: 1, 5, 40, 385, 4095, 46376, 548340, 6690585, 83615350, 1064887395, 13770292256, 

[1, 3, 3]: 1, 6, 57, 650, 8184, 109668, 1533939, 22137570, 327203085, 4928006512, 75357373305, 

[1, 3, 4]: 1, 7, 77, 1015, 14763, 228459, 3689595, 61474519, 1048927880, 18236463245, 321899509386, 

[1, 3, 5]: 1, 8, 100, 1496, 24682, 433160, 7932196, 149846840, 2898753715, 57135036024, 1143315429776, 

[1, 4, -5]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 

[1, 4, -4]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

[1, 4, -3]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[1, 4, -2]: 1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 

[1, 4, -1]: 1, 3, 15, 91, 612, 4389, 32890, 254475, 2017356, 16301164, 133767543, 

[1, 4, 0]: 1, 4, 26, 204, 1771, 16380, 158224, 1577532, 16112057, 167710664, 1772645420, 

[1, 4, 1]: 1, 5, 40, 385, 4095, 46376, 548340, 6690585, 83615350, 1064887395, 13770292256, 

[1, 4, 2]: 1, 6, 57, 650, 8184, 109668, 1533939, 22137570, 327203085, 4928006512, 75357373305, 

[1, 4, 3]: 1, 7, 77, 1015, 14763, 228459, 3689595, 61474519, 1048927880, 18236463245, 321899509386, 

[1, 4, 4]: 1, 8, 100, 1496, 24682, 433160, 7932196, 149846840, 2898753715, 57135036024, 1143315429776, 

[1, 4, 5]: 1, 9, 126, 2109, 38916, 763686, 15636192, 330237765, 7141879503, 157366449604, 3520256293710, 

[1, 5, -5]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

[1, 5, -4]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[1, 5, -3]: 1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 

[1, 5, -2]: 1, 3, 15, 91, 612, 4389, 32890, 254475, 2017356, 16301164, 133767543, 

[1, 5, -1]: 1, 4, 26, 204, 1771, 16380, 158224, 1577532, 16112057, 167710664, 1772645420, 

[1, 5, 0]: 1, 5, 40, 385, 4095, 46376, 548340, 6690585, 83615350, 1064887395, 13770292256, 

[1, 5, 1]: 1, 6, 57, 650, 8184, 109668, 1533939, 22137570, 327203085, 4928006512, 75357373305, 

[1, 5, 2]: 1, 7, 77, 1015, 14763, 228459, 3689595, 61474519, 1048927880, 18236463245, 321899509386, 

[1, 5, 3]: 1, 8, 100, 1496, 24682, 433160, 7932196, 149846840, 2898753715, 57135036024, 1143315429776, 

[1, 5, 4]: 1, 9, 126, 2109, 38916, 763686, 15636192, 330237765, 7141879503, 157366449604, 3520256293710, 

[1, 5, 5]: 1, 10, 155, 2870, 58565, 1270752, 28765650, 671650110, 16057800980, 391139588190, 9672348219898, 

[2, -5, -5]: 1, -5, 30, -185, 1090, -5626, 19720, 45695, -1929840, 26595030, -282947496, 

[2, -5, -4]: 1, -4, 17, -60, 79, 1296, -17360, 142068, -873895, 3711532, -2373973, 

[2, -5, -3]: 1, -3, 7, 5, -192, 1347, -5258, 1557, 160270, -1442000, 6975729, 

[2, -5, -2]: 1, -2, 0, 26, -123, 98, 2224, -14430, 23967, 245364, -2067680, 

[2, -5, -1]: 1, -1, -4, 19, 11, -336, 676, 4699, -26764, -25371, 663520, 

[2, -5, 0]: 1, 0, -5, 0, 60, 0, -935, 0, 16555, 0, -316251, 

[2, -5, 1]: 1, 1, -3, -15, -1, 197, 527, -1711, -13530, -7993, 221780, 

[2, -5, 2]: 1, 2, 2, -10, -72, -222, -34, 3598, 20735, 50630, -109596, 

[2, -5, 3]: 1, 3, 10, 31, 72, -6, -1540, -13065, -78479, -381576, -1474818, 

[2, -5, 4]: 1, 4, 21, 124, 781, 5120, 34474, 236492, 1644222, 11543644, 81623504, 

[2, -5, 5]: 1, 5, 35, 285, 2530, 23751, 231880, 2330445, 23950355, 250543370, 2658968130, 

[2, -4, -5]: 1, -5, 31, -205, 1368, -8891, 54102, -286173, 1020910, 2638800, -110587783, 

[2, -4, -4]: 1, -4, 18, -76, 247, -92, -8136, 97092, -798813, 5301608, -28151584, 

[2, -4, -3]: 1, -3, 8, -7, -109, 1008, -5316, 16145, 19220, -665873, 5278504, 

[2, -4, -2]: 1, -2, 1, 18, -100, 196, 871, -8758, 29971, 27512, -858793, 

[2, -4, -1]: 1, -1, -3, 15, -1, -197, 527, 1711, -13530, 7993, 221780, 

[2, -4, 0]: 1, 0, -4, 0, 38, 0, -468, 0, 6545, 0, -98728, 

[2, -4, 1]: 1, 1, -2, -11, -8, 94, 336, -259, -5087, -10712, 41026, 

[2, -4, 2]: 1, 2, 3, -2, -39, -176, -442, -26, 6222, 36062, 113240, 

[2, -4, 3]: 1, 3, 11, 43, 170, 657, 2392, 7675, 17603, -11898, -529678, 

[2, -4, 4]: 1, 4, 22, 140, 969, 7084, 53820, 420732, 3362260, 27343888, 225568798, 

[2, -4, 5]: 1, 5, 36, 305, 2833, 27916, 286632, 3033513, 32858595, 362515725, 4059475368, 

[2, -3, -5]: 1, -5, 32, -225, 1651, -12336, 92460, -687161, 5006180, -35235655, 234112824, 

[2, -3, -4]: 1, -4, 19, -92, 420, -1624, 3545, 20372, -386621, 3865104, -31129539, 

[2, -3, -3]: 1, -3, 9, -19, -21, 561, -4093, 20493, -69030, 43819, 1571700, 

[2, -3, -2]: 1, -2, 2, 10, -72, 222, -34, -3598, 20735, -50630, -109596, 

[2, -3, -1]: 1, -1, -2, 11, -8, -94, 336, 259, -5087, 10712, 41026, 

[2, -3, 0]: 1, 0, -3, 0, 21, 0, -190, 0, 1950, 0, -21576, 

[2, -3, 1]: 1, 1, -1, -7, -10, 27, 152, 169, -949, -4286, -2646, 

[2, -3, 2]: 1, 2, 4, 6, -1, -58, -304, -1090, -2876, -4216, 9244, 

[2, -3, 3]: 1, 3, 12, 55, 273, 1428, 7752, 43263, 246675, 1430715, 8414640, 

[2, -3, 4]: 1, 4, 23, 156, 1162, 9192, 75819, 644908, 5616182, 49826712, 448771622, 

[2, -3, 5]: 1, 5, 37, 325, 3141, 32261, 345605, 3818501, 43197445, 497868805, 5825331205, 

[2, -2, -5]: 1, -5, 33, -245, 1939, -15961, 134843, -1159829, 10103730, -88820835, 785856036, 

[2, -2, -4]: 1, -4, 20, -108, 598, -3300, 17732, -90140, 411281, -1451980, 965192, 

[2, -2, -3]: 1, -3, 10, -31, 72, 6, -1540, 13065, -78479, 381576, -1474818, 

[2, -2, -2]: 1, -2, 3, 2, -39, 176, -442, 26, 6222, -36062, 113240, 

[2, -2, -1]: 1, -1, -1, 7, -10, -27, 152, -169, -949, 4286, -2646, 

[2, -2, 0]: 1, 0, -2, 0, 9, 0, -52, 0, 340, 0, -2394, 

[2, -2, 1]: 1, 1, 0, -3, -7, -4, 24, 85, 99, -215, -1196, 

[2, -2, 2]: 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 

[2, -2, 3]: 1, 3, 13, 67, 381, 2307, 14589, 95235, 636925, 4341763, 30056445, 

[2, -2, 4]: 1, 4, 24, 172, 1360, 11444, 100520, 911068, 8457504, 80006116, 768464312, 

[2, -2, 5]: 1, 5, 38, 345, 3454, 36786, 408848, 4687969, 55048310, 658645110, 8001060132, 

[2, -1, -5]: 1, -5, 34, -265, 2232, -19766, 181300, -1706737, 16392049, -159959240, 1581278838, 

[2, -1, -4]: 1, -4, 21, -124, 781, -5120, 34474, -236492, 1644222, -11543644, 81623504, 

[2, -1, -3]: 1, -3, 11, -43, 170, -657, 2392, -7675, 17603, 11898, -529678, 

[2, -1, -2]: 1, -2, 4, -6, -1, 58, -304, 1090, -2876, 4216, 9244, 

[2, -1, -1]: 1, -1, 0, 3, -7, 4, 24, -85, 99, 215, -1196, 

[2, -1, 0]: 1, 0, -1, 0, 2, 0, -5, 0, 14, 0, -42, 

[2, -1, 1]: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 

[2, -1, 2]: 1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 

[2, -1, 3]: 1, 3, 14, 79, 494, 3294, 22952, 165127, 1217270, 9146746, 69799476, 

[2, -1, 4]: 1, 4, 25, 188, 1563, 13840, 127972, 1221260, 11938471, 118936100, 1203155633, 

[2, -1, 5]: 1, 5, 39, 365, 3772, 41491, 476410, 5644477, 68493324, 846937140, 10633195119, 

[2, 0, -5]: 1, -5, 35, -285, 2530, -23751, 231880, -2330445, 23950355, -250543370, 2658968130, 

[2, 0, -4]: 1, -4, 22, -140, 969, -7084, 53820, -420732, 3362260, -27343888, 225568798, 

[2, 0, -3]: 1, -3, 12, -55, 273, -1428, 7752, -43263, 246675, -1430715, 8414640, 

[2, 0, -2]: 1, -2, 5, -14, 42, -132, 429, -1430, 4862, -16796, 58786, 

[2, 0, -1]: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 

[2, 0, 0]: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

[2, 0, 1]: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 

[2, 0, 2]: 1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 

[2, 0, 3]: 1, 3, 15, 91, 612, 4389, 32890, 254475, 2017356, 16301164, 133767543, 

[2, 0, 4]: 1, 4, 26, 204, 1771, 16380, 158224, 1577532, 16112057, 167710664, 1772645420, 

[2, 0, 5]: 1, 5, 40, 385, 4095, 46376, 548340, 6690585, 83615350, 1064887395, 13770292256, 

[2, 1, -5]: 1, -5, 36, -305, 2833, -27916, 286632, -3033513, 32858595, -362515725, 4059475368, 

[2, 1, -4]: 1, -4, 23, -156, 1162, -9192, 75819, -644908, 5616182, -49826712, 448771622, 

[2, 1, -3]: 1, -3, 13, -67, 381, -2307, 14589, -95235, 636925, -4341763, 30056445, 

[2, 1, -2]: 1, -2, 6, -22, 90, -394, 1806, -8558, 41586, -206098, 1037718, 

[2, 1, -1]: 1, -1, 2, -5, 14, -42, 132, -429, 1430, -4862, 16796, 

[2, 1, 0]: 1, 0, 1, 0, 3, 0, 12, 0, 55, 0, 273, 

[2, 1, 1]: 1, 1, 3, 9, 32, 119, 466, 1881, 7788, 32868, 140907, 

[2, 1, 2]: 1, 2, 8, 38, 201, 1134, 6688, 40734, 254237, 1617572, 10452416, 

[2, 1, 3]: 1, 3, 16, 103, 735, 5592, 44452, 364815, 3067558, 26290517, 228819168, 

[2, 1, 4]: 1, 4, 27, 220, 1984, 19064, 191325, 1981932, 21031965, 227463808, 2498039219, 

[2, 1, 5]: 1, 5, 41, 405, 4423, 51441, 624687, 7828853, 100497980, 1314688375, 17462945394, 

[2, 2, -5]: 1, -5, 37, -325, 3141, -32261, 345605, -3818501, 43197445, -497868805, 5825331205, 

[2, 2, -4]: 1, -4, 24, -172, 1360, -11444, 100520, -911068, 8457504, -80006116, 768464312, 

[2, 2, -3]: 1, -3, 14, -79, 494, -3294, 22952, -165127, 1217270, -9146746, 69799476, 

[2, 2, -2]: 1, -2, 7, -30, 143, -728, 3876, -21318, 120175, -690690, 4032015, 

[2, 2, -1]: 1, -1, 3, -9, 32, -119, 466, -1881, 7788, -32868, 140907, 

[2, 2, 0]: 1, 0, 2, 0, 11, 0, 80, 0, 665, 0, 5980, 

[2, 2, 1]: 1, 1, 4, 13, 55, 232, 1052, 4869, 23206, 112519, 554560, 

[2, 2, 2]: 1, 2, 9, 46, 264, 1612, 10291, 67830, 458109, 3153744, 22049065, 

[2, 2, 3]: 1, 3, 17, 115, 863, 6903, 57687, 497683, 4398980, 39630305, 362562226, 

[2, 2, 4]: 1, 4, 28, 236, 2202, 21892, 227324, 2436508, 26752627, 299369532, 3401762928, 

[2, 2, 5]: 1, 5, 42, 425, 4756, 56686, 705500, 9061841, 119225535, 1598582580, 21763800786, 

[2, 3, -5]: 1, -5, 38, -345, 3454, -36786, 408848, -4687969, 55048310, -658645110, 8001060132, 

[2, 3, -4]: 1, -4, 25, -188, 1563, -13840, 127972, -1221260, 11938471, -118936100, 1203155633, 

[2, 3, -3]: 1, -3, 15, -91, 612, -4389, 32890, -254475, 2017356, -16301164, 133767543, 

[2, 3, -2]: 1, -2, 8, -38, 201, -1134, 6688, -40734, 254237, -1617572, 10452416, 

[2, 3, -1]: 1, -1, 4, -13, 55, -232, 1052, -4869, 23206, -112519, 554560, 

[2, 3, 0]: 1, 0, 3, 0, 24, 0, 253, 0, 3045, 0, 39627, 

[2, 3, 1]: 1, 1, 5, 17, 83, 381, 1939, 9905, 52544, 282315, 1545130, 

[2, 3, 2]: 1, 2, 10, 54, 332, 2162, 14734, 103630, 746857, 5486206, 40926152, 

[2, 3, 3]: 1, 3, 18, 127, 996, 8322, 72644, 654615, 6043455, 56866028, 543368586, 

[2, 3, 4]: 1, 4, 29, 252, 2425, 24864, 266270, 2943308, 33329204, 384641836, 4507577438, 

[2, 3, 5]: 1, 5, 43, 445, 5094, 62111, 790828, 10392109, 139883065, 1918862510, 26727571728, 

[2, 4, -5]: 1, -5, 39, -365, 3772, -41491, 476410, -5644477, 68493324, -846937140, 10633195119, 

[2, 4, -4]: 1, -4, 26, -204, 1771, -16380, 158224, -1577532, 16112057, -167710664, 1772645420, 

[2, 4, -3]: 1, -3, 16, -103, 735, -5592, 44452, -364815, 3067558, -26290517, 228819168, 

[2, 4, -2]: 1, -2, 9, -46, 264, -1612, 10291, -67830, 458109, -3153744, 22049065, 

[2, 4, -1]: 1, -1, 5, -17, 83, -381, 1939, -9905, 52544, -282315, 1545130, 

[2, 4, 0]: 1, 0, 4, 0, 42, 0, 580, 0, 9139, 0, 155664, 

[2, 4, 1]: 1, 1, 6, 21, 116, 566, 3176, 17501, 101391, 590756, 3519782, 

[2, 4, 2]: 1, 2, 11, 62, 405, 2784, 20066, 149158, 1136276, 8821958, 69556554, 

[2, 4, 3]: 1, 3, 19, 139, 1134, 9849, 89372, 837147, 8033545, 78573186, 780388752, 

[2, 4, 4]: 1, 4, 30, 268, 2653, 27980, 308212, 3504380, 40817586, 484534720, 5840593274, 

[2, 4, 5]: 1, 5, 44, 465, 5437, 67716, 880720, 11822217, 162556349, 2277870665, 32411053200, 

[2, 5, -5]: 1, -5, 40, -385, 4095, -46376, 548340, -6690585, 83615350, -1064887395, 13770292256, 

[2, 5, -4]: 1, -4, 27, -220, 1984, -19064, 191325, -1981932, 21031965, -227463808, 2498039219, 

[2, 5, -3]: 1, -3, 17, -115, 863, -6903, 57687, -497683, 4398980, -39630305, 362562226, 

[2, 5, -2]: 1, -2, 10, -54, 332, -2162, 14734, -103630, 746857, -5486206, 40926152, 

[2, 5, -1]: 1, -1, 6, -21, 116, -566, 3176, -17501, 101391, -590756, 3519782, 

[2, 5, 0]: 1, 0, 5, 0, 65, 0, 1110, 0, 21620, 0, 455126, 

[2, 5, 1]: 1, 1, 7, 25, 154, 787, 4812, 28169, 176065, 1096342, 7020112, 

[2, 5, 2]: 1, 2, 12, 70, 483, 3478, 26336, 205438, 1642890, 13388000, 110796476, 

[2, 5, 3]: 1, 3, 20, 151, 1277, 11484, 107920, 1046815, 10402541, 105357279, 1083566504, 

[2, 5, 4]: 1, 4, 31, 284, 2886, 31240, 353199, 4121772, 49274392, 600342184, 7427285236, 

[2, 5, 5]: 1, 5, 45, 485, 5785, 73501, 975225, 13354725, 187331895, 2677999545, 38873136507,