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A296044 a(n) = [x^n] Product_{k>=1} ((1 + x^(2*k))/(1 - x^(2*k-1)))^n.

%I A296044 #16 Jan 17 2024 08:18:23
%S A296044 1,1,5,22,101,481,2330,11425,56549,281911,1413465,7120136,36006362,
%T A296044 182681916,929461993,4740491107,24229115109,124069449335,636376573943,
%U A296044 3268955179686,16814509004601,86593280920756,446437797872016,2303948443259841,11900990745759578,61526182236027756
%N A296044 a(n) = [x^n] Product_{k>=1} ((1 + x^(2*k))/(1 - x^(2*k-1)))^n.
%H A296044 G. C. Greubel, <a href="/A296044/b296044.txt">Table of n, a(n) for n = 0..500</a>
%F A296044 a(n) = [x^n] Product_{k>=1} ((1 - x^(4*k))/(1 - x^k))^n.
%F A296044 a(n) ~ c * d^n / sqrt(n), where d = 5.2749356339591798618290252741994029798069148326559... and c = 0.2726256757090475625917361066565981461455343437... - _Vaclav Kotesovec_, Dec 05 2017
%t A296044 Table[SeriesCoefficient[Product[((1 + x^(2 k))/(1 - x^(2 k - 1)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
%t A296044 Table[SeriesCoefficient[Product[((1 - x^(4 k))/(1 - x^k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
%t A296044 Table[SeriesCoefficient[(EllipticTheta[2, 0, x]/EllipticTheta[2, Pi/4, x^(1/2)]/(16 x)^(1/8))^n, {x, 0, n}], {n, 0, 25}]
%t A296044 (* Calculation of constant d: *) With[{k = 4}, 1/r /. FindRoot[{s == QPochhammer[(r*s)^k] / QPochhammer[r*s], k*(-(s*QPochhammer[r*s]*(Log[1 - (r*s)^k] + QPolyGamma[0, 1, (r*s)^k]) / Log[(r*s)^k]) + (r*s)^k * Derivative[0, 1][QPochhammer][(r*s)^k, (r*s)^k]) == s*QPochhammer[r*s] + s^2*(-(QPochhammer[r*s]*(Log[1 - r*s] + QPolyGamma[0, 1, r*s]) / (s*Log[r*s])) + r*Derivative[0, 1][QPochhammer][r*s, r*s])}, {r, 1/5}, {s, 1}, WorkingPrecision -> 70]] (* _Vaclav Kotesovec_, Jan 17 2024 *)
%Y A296044 Main diagonal of A296068.
%Y A296044 Cf. A001935, A001936, A001937, A001939, A001940, A001941, A092877, A093160, A295831, A296043, A296045.
%K A296044 nonn
%O A296044 0,3
%A A296044 _Ilya Gutkovskiy_, Dec 03 2017