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A385089 G.f.: Sum_{k>=0} x^k * Product_{j=1..3*k} (1 + x^j)/(1 - x^j).

%I A385089 #9 Jun 18 2025 07:34:40
%S A385089 1,1,3,7,15,27,47,79,127,199,307,465,695,1025,1493,2151,3069,4337,
%T A385089 6075,8441,11639,15933,21667,29281,39337,52555,69849,92375,121595,
%U A385089 159347,207939,270259,349911,451377,580223,743341,949241,1208415,1533763,1941111,2449841,3083637
%N A385089 G.f.: Sum_{k>=0} x^k * Product_{j=1..3*k} (1 + x^j)/(1 - x^j).
%H A385089 Vaclav Kotesovec, <a href="/A385089/b385089.txt">Table of n, a(n) for n = 0..5000</a>
%F A385089 a(n) ~ Gamma(1/3) * exp(Pi*sqrt(n)) / (3 * 2^(8/3) * Pi^(2/3) * n^(2/3)).
%t A385089 nmax = 50; CoefficientList[Series[Sum[x^k*Product[(1+x^j)/(1-x^j), {j, 1, 3*k}], {k, 0, nmax}], {x, 0, nmax}], x]
%t A385089 nmax = 50; p = 1; q = 1; s = 1; Do[p = Expand[p*(1 - x^(3*k))*(1 - x^(3*k - 1))*(1 - x^(3*k - 2))]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; q = Expand[q*(1 + x^(3*k))*(1 + x^(3*k - 1))*(1 + x^(3*k - 2))]; 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 A385089 Cf. A035295, A385067.
%Y A385089 Cf. A207641, A385088, A385090, A385091, A385092.
%K A385089 nonn
%O A385089 0,3
%A A385089 _Vaclav Kotesovec_, Jun 17 2025