OFFSET
0,3
COMMENTS
In general, if m > 1 and g.f. = Product_{k>=1} (1 - x^(m*k))^(m*k)/((1 - x^k)^k), then a(n, m) ~ exp(1/12 - m/12 + 3 * 2^(-2/3) * (1-1/m)^(1/3) * Zeta(3)^(1/3) * n^(2/3)) * 2^(-(m+11)/36) * A^(m-1) * (m-1)^((7-m)/36) * m^(-(2*m+7)/36) * Zeta(3)^((7-m)/36) * n^((m-25)/36) / sqrt(3*Pi), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Apr 16 2017
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000
FORMULA
G.f.: Product_{k>=0} 1 / ((1-x^(5*k+1))^(5*k+1) * (1-x^(5*k+2))^(5*k+2) * (1-x^(5*k+3))^(5*k+3) * (1-x^(5*k+4))^(5*k+4)).
a(n) ~ exp(-1/3 + 3*(Zeta(3)/5)^(1/3)*n^(2/3)) * A^4 * Zeta(3)^(1/18) / (2^(1/3) * 5^(17/36) * sqrt(3*Pi) * n^(5/9)), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Apr 16 2017
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[1 / ((1-x^(5*k+1))^(5*k+1) * (1-x^(5*k+2))^(5*k+2) * (1-x^(5*k+3))^(5*k+3) * (1-x^(5*k+4))^(5*k+4)), {k, 0, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Apr 15 2017 *)
nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))^(5*k)/((1 - x^k)^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Apr 15 2017 *)
PROG
(PARI) x='x+O('x^100); Vec(prod(k=0, 100, 1 / ((1 - x^(5*k + 1))^(5*k + 1)*(1 - x^(5*k + 2))^(5*k + 2)*(1 - x^(5*k + 3))^(5*k + 3)*(1 - x^(5*k + 4))^(5*k + 4)))) \\ _Indranil Ghosh_, Apr 15 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
_Seiichi Manyama_, Apr 15 2017
STATUS
approved