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Expansion of Product_{k>=0} (1-x^(5*k+1))^(5*k+1).
+10
5
1, -1, 0, 0, 0, 0, -6, 6, 0, 0, 0, -11, 26, -15, 0, 0, -16, 82, -86, 20, 0, -21, 172, -316, 180, -15, -26, 328, -872, 790, -226, -25, 538, -2043, 2681, -1310, 130, 843, -4184, 7426, -5390, 1365, 1158, -7855, 18067, -17705, 7185, 798, -13701, 39468, -50030
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[(1-x^(5*k-4))^(5*k-4), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 17 2017 *)
Expansion of Product_{k>=0} 1/(1-x^(4*k+1))^(4*k+1).
+10
4
1, 1, 1, 1, 1, 6, 6, 6, 6, 15, 30, 30, 30, 43, 88, 123, 123, 140, 250, 385, 455, 476, 678, 1098, 1413, 1564, 1913, 2918, 4048, 4707, 5452, 7572, 10747, 13265, 15195, 19534, 27349, 35146, 41042, 50011, 67596, 88897, 106519, 126635, 164230, 216862, 266473, 314883
FORMULA
a(n) ~ 4 * Pi * 2^(25/72) * Zeta(3)^(11/72) * exp(4*c + 3 * 2^(-4/3) * Zeta(3)^(1/3) * n^(2/3)) / (sqrt(3) * Gamma(1/4)^3 * n^(47/72)), where c = Integral_{x=0..inf} ((-19/(exp(x)*96) + 1/(exp(x)*(1 - exp(-4*x))^2) - 1/(16*x^2) - 3/(16*x))/x) dx = 0.09601010361866957956805888476415949391295401812706635... - Vaclav Kotesovec, Apr 16 2017
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[1/(1-x^(4*k-3))^(4*k-3), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 16 2017 *)
CROSSREFS
Product_{k>=0} 1/(1-x^(m*k+1))^(m*k+1): A262811 (m=2), A262947 (m=3), this sequence (m=4), A285049 (m=5).
Expansion of Product_{k>=0} (1-x^(3*k+1))^(3*k+1).
+10
4
1, -1, 0, 0, -4, 4, 0, -7, 13, -6, -10, 38, -32, -9, 74, -103, 27, 137, -266, 153, 191, -593, 537, 167, -1161, 1437, -222, -2035, 3397, -1578, -3110, 7160, -5285, -3712, 13942, -13920, -2002, 24848, -32241, 6764, 40661, -68059, 32487, 59109, -133506, 95221, 71243
Expansion of Product_{k>=0} (1 + x^(4*k+1))^(4*k+1).
+10
4
1, 1, 0, 0, 0, 5, 5, 0, 0, 9, 19, 10, 0, 13, 58, 55, 10, 17, 118, 191, 95, 26, 223, 512, 400, 116, 362, 1175, 1329, 564, 609, 2368, 3593, 2218, 1246, 4402, 8600, 7118, 3433, 7792, 18503, 19778, 10702, 13924, 37009, 49017, 32097, 27141, 69629, 111251, 88972
FORMULA
a(n) ~ exp(3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / 4) * Zeta(3)^(1/6) / (2^(23/24) * 3^(1/3) * sqrt(Pi) * n^(2/3)). - Vaclav Kotesovec, Apr 16 2017
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[(1 + x^(4*k-3))^(4*k-3), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 16 2017 *)
CROSSREFS
Product_{k>=0} (1 + x^(m*k+1))^(m*k+1): A262736 (m=2), A262949 (m=3), this sequence (m=4).
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