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%I #27 Apr 16 2017 07:33:31
%S 1,1,3,6,9,20,36,62,106,184,302,503,829,1325,2119,3367,5282,8227,
%T 12740,19550,29849,45300,68325,102495,152998,227249,336005,494597,
%U 724875,1058213,1538860,2229370,3218304,4630015,6638728,9488894,13520995,19208916,27211430
%N Expansion of Product_{k>=1} (1 - x^(4*k))^(4*k) / (1 - x^k)^k.
%H Seiichi Manyama, <a href="/A285215/b285215.txt">Table of n, a(n) for n = 0..10000</a>
%F G.f.: Product_{k>=0} 1 / ((1-x^(4*k+1))^(4*k+1) * (1-x^(4*k+2))^(4*k+2) * (1-x^(4*k+3))^(4*k+3)).
%F a(n) ~ exp(-1/4 + 2^(-4/3) * 3^(4/3) * Zeta(3)^(1/3) * n^(2/3)) * A^3 * Zeta(3)^(1/12) / (2^(5/4) * 3^(5/12) * sqrt(Pi) * n^(7/12)), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Apr 16 2017
%t nmax = 50; CoefficientList[Series[Product[1 / ((1-x^(4*k+1))^(4*k+1) * (1-x^(4*k+2))^(4*k+2) * (1-x^(4*k+3))^(4*k+3)), {k,0,nmax}], {x,0,nmax}], x] (* _Vaclav Kotesovec_, Apr 15 2017 *)
%t nmax = 50; CoefficientList[Series[Product[(1 - x^(4*k))^(4*k)/((1 - x^k)^k), {k,1,nmax}], {x,0,nmax}], x] (* _Vaclav Kotesovec_, Apr 15 2017 *)
%o (PARI) x='x+O('x^100); Vec(prod(k=0, 100, 1 / ((1 - x^(4*k + 1))^(4*k + 1)*(1 - x^(4*k + 2))^(4*k + 2)*(1 - x^(4*k + 3))^(4*k + 3)))) \\ _Indranil Ghosh_, Apr 15 2017
%Y Product_{k>=1} (1 - x^(m*k))^(m*k)/(1 - x^k)^k: A262811 (m=2), A262923 (m=3), this sequence (m=4), A285246 (m=5).
%Y Cf. A285262, A285284.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Apr 15 2017
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