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A385090 G.f.: Sum_{k>=0} x^k * Product_{j=1..4*k} (1 + x^j)/(1 - x^j).

%I A385090 #9 Jun 18 2025 07:34:26
%S A385090 1,1,3,7,15,29,51,87,143,227,353,537,803,1185,1727,2489,3551,5021,
%T A385090 7039,9791,13521,18541,25261,34207,46051,61655,82113,108815,143517,
%U A385090 188433,246343,320725,415931,537377,691791,887517,1134863,1446549,1838235,2329147,2942849,3708165
%N A385090 G.f.: Sum_{k>=0} x^k * Product_{j=1..4*k} (1 + x^j)/(1 - x^j).
%H A385090 Vaclav Kotesovec, <a href="/A385090/b385090.txt">Table of n, a(n) for n = 0..5000</a>
%F A385090 a(n) ~ Gamma(1/4) * exp(Pi*sqrt(n)) / (2^(9/2) * Pi^(3/4) * n^(5/8)).
%t A385090 nmax = 50; CoefficientList[Series[Sum[x^k*Product[(1+x^j)/(1-x^j), {j, 1, 4*k}], {k, 0, nmax}], {x, 0, nmax}], x]
%t A385090 nmax = 50; p = 1; q = 1; s = 1; Do[p = Expand[p*(1 - x^(4*k))*(1 - x^(4*k - 1))*(1 - x^(4*k - 2))*(1 - x^(4*k - 3))]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; q = Expand[q*(1 + x^(4*k))*(1 + x^(4*k - 1))*(1 + x^(4*k - 2))*(1 + x^(4*k - 3))]; q = Take[q, Min[nmax + 1, Exponent[q, x] + 1, Length[q]]]; s += x^k*q/p;, {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x]
%Y A385090 Cf. A035296, A385068.
%Y A385090 Cf. A207641, A385088, A385089, A385091, A385092.
%K A385090 nonn
%O A385090 0,3
%A A385090 _Vaclav Kotesovec_, Jun 17 2025