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A304629 a(n) = [x^n] Product_{k>=1} ((1 + x^k)/(1 + x^(5*k)))^n.

%I A304629 #14 Jan 17 2024 07:18:59
%S A304629 1,1,3,13,51,201,819,3389,14131,59341,250703,1064207,4535091,19390229,
%T A304629 83139955,357354213,1539272499,6642769925,28714955571,124312591469,
%U A304629 538895612751,2338948779320,10162837993377,44202371860240,192431323820851,838442649862701,3656031108325651
%N A304629 a(n) = [x^n] Product_{k>=1} ((1 + x^k)/(1 + x^(5*k)))^n.
%H A304629 G. C. Greubel, <a href="/A304629/b304629.txt">Table of n, a(n) for n = 0..500</a>
%F A304629 a(n) = [x^n] Product_{k>=1} 1/(1 - x^k + x^(2*k) - x^(3*k) + x^(4*k))^n.
%F A304629 a(n) ~ c * d^n / sqrt(n), where d = 4.445766346387064439086120427... and c = 0.267035948020079842478289... - _Vaclav Kotesovec_, May 18 2018
%t A304629 Table[SeriesCoefficient[Product[((1 + x^k)/(1 + x^(5 k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
%t A304629 Table[SeriesCoefficient[Product[1/(1 - x^k + x^(2 k) - x^(3 k) + x^(4 k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
%t A304629 (* Calculation of constants {d,c}: *) With[{k = 5}, {1/r, Sqrt[QPochhammer[-1, (r*s)^k] / (2*Pi*(r^2*s*Derivative[0, 2][QPochhammer][-1, r*s] - k^2*(r*s)^(2*k) * Derivative[0, 2][QPochhammer][-1, (r*s)^k] - k*(1 + k)*(r*s)^k * Derivative[0, 1][QPochhammer][-1, (r*s)^k]))]} /. FindRoot[{s == QPochhammer[-1, r*s] / QPochhammer[-1, (r*s)^k], QPochhammer[-1, (r*s)^k] + k*(r*s)^k*Derivative[0, 1][QPochhammer][-1, (r*s)^k] == r*Derivative[0, 1][QPochhammer][-1, r*s]}, {r, 1/4}, {s, 2}, WorkingPrecision -> 70]] (* _Vaclav Kotesovec_, Jan 17 2024 *)
%Y A304629 Cf. A096938, A255526, A285291, A296163, A296164, A304628.
%K A304629 nonn
%O A304629 0,3
%A A304629 _Ilya Gutkovskiy_, May 15 2018