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a(n) = [x^n] Product_{k>=1} ((1 + x^k)/(1 + x^(4*k)))^n.
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%I #12 Jan 17 2024 07:21:23

%S 1,1,3,13,47,181,729,2948,12031,49540,205153,853546,3565505,14943839,

%T 62810786,264650683,1117486463,4727486583,20032950744,85017558081,

%U 361289789377,1537198394570,6547611493822,27917246924099,119141276756545,508884954441331,2175284934712217,9305217981192748

%N a(n) = [x^n] Product_{k>=1} ((1 + x^k)/(1 + x^(4*k)))^n.

%H G. C. Greubel, <a href="/A304628/b304628.txt">Table of n, a(n) for n = 0..500</a>

%F a(n) = [x^n] Product_{k>=1} ((1 - x^(8*k-4))/(1 - x^(2*k-1)))^n.

%F a(n) ~ c * d^n / sqrt(n), where d = 4.3582188263213968630940316689... and c = 0.266443662680498334500839... - _Vaclav Kotesovec_, May 18 2018

%t Table[SeriesCoefficient[Product[((1 + x^k)/(1 + x^(4 k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 27}]

%t Table[SeriesCoefficient[Product[((1 - x^(8 k - 4))/(1 - x^(2 k - 1)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 27}]

%t (* Calculation of constants {d,c}: *) With[{k = 4}, {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 Cf. A070048, A255526, A285290, A296044, A296164, A304629.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, May 15 2018