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A385091 G.f.: Sum_{k>=0} x^k * Product_{j=1..5*k} (1 + x^j)/(1 - x^j).

%I A385091 #10 Jun 18 2025 07:41:28
%S A385091 1,1,3,7,15,29,53,91,151,243,381,585,881,1305,1907,2753,3931,5559,
%T A385091 7793,10835,14955,20501,27921,37801,50889,68139,90777,120353,158827,
%U A385091 208683,273037,355791,461839,597273,769661,988411,1265149,1614215,2053297,2604113,3293281,4153407
%N A385091 G.f.: Sum_{k>=0} x^k * Product_{j=1..5*k} (1 + x^j)/(1 - x^j).
%H A385091 Vaclav Kotesovec, <a href="/A385091/b385091.txt">Table of n, a(n) for n = 0..5000</a>
%F A385091 a(n) ~ Gamma(1/5) * exp(Pi*sqrt(n)) / (5 * 2^(12/5) * Pi^(4/5) * n^(3/5)).
%t A385091 nmax = 50; CoefficientList[Series[Sum[x^k*Product[(1+x^j)/(1-x^j), {j, 1, 5*k}], {k, 0, nmax}], {x, 0, nmax}], x]
%t A385091 nmax = 50; p = 1; q = 1; s = 1; Do[p = Expand[p*(1 - x^(5*k))*(1 - x^(5*k - 1))*(1 - x^(5*k - 2))*(1 - x^(5*k - 3))*(1 - x^(5*k - 4))]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; q = Expand[q*(1 + x^(5*k))*(1 + x^(5*k - 1))*(1 + x^(5*k - 2))*(1 + x^(5*k - 3))*(1 + x^(5*k - 4))]; 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 A385091 Cf. A035297, A385069.
%Y A385091 Cf. A207641, A385088, A385089, A385090, A385092.
%K A385091 nonn
%O A385091 0,3
%A A385091 _Vaclav Kotesovec_, Jun 17 2025