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A285246 Expansion of Product_{k>=1} (1 - x^(5*k))^(5*k) / (1 - x^k)^k.

%I A285246 #25 Apr 17 2017 13:05:16
%S A285246 1,1,3,6,13,19,43,71,130,217,380,619,1049,1685,2757,4404,7027,11014,
%T A285246 17326,26820,41488,63514,96970,146808,221659,332212,496439,737535,
%U A285246 1091938,1608564,2361929,3452736,5031138,7302373,10566038,15234196,21900182,31380435
%N A285246 Expansion of Product_{k>=1} (1 - x^(5*k))^(5*k) / (1 - x^k)^k.
%C A285246 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
%H A285246 Seiichi Manyama, <a href="/A285246/b285246.txt">Table of n, a(n) for n = 0..10000</a>
%F A285246 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)).
%F A285246 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
%t A285246 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 *)
%t A285246 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 *)
%o A285246 (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
%Y A285246 Product_{k>=1} (1 - x^(m*k))^(m*k)/(1 - x^k)^k: A262811 (m=2), A262923 (m=3), A285215 (m=4), this sequence (m=5).
%Y A285246 Cf. A285263, A285285.
%K A285246 nonn
%O A285246 0,3
%A A285246 _Seiichi Manyama_, Apr 15 2017