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A385092 G.f.: Sum_{k>=0} x^k * Product_{j=1..6*k} (1 + x^j)/(1 - x^j).

%I A385092 #11 Jun 18 2025 07:41:53
%S A385092 1,1,3,7,15,29,53,93,155,251,397,613,929,1385,2033,2945,4219,5979,
%T A385092 8393,11683,16133,22119,30125,40773,54867,73435,97785,129583,170941,
%U A385092 224519,293673,382615,496609,642231,827667,1063073,1361029,1737081,2210381,2804485,3548303,4477229
%N A385092 G.f.: Sum_{k>=0} x^k * Product_{j=1..6*k} (1 + x^j)/(1 - x^j).
%C A385092 In general, for m>=1, if g.f. = Sum_{k>=0} x^k * Product_{j=1..m*k} (1 + x^j)/(1 - x^j), then a(n) ~ Gamma(1/m) * exp(Pi*sqrt(n)) / (m * 2^(2 + 2/m) * Pi^(1 - 1/m) * n^((m+1)/(2*m))).
%H A385092 Vaclav Kotesovec, <a href="/A385092/b385092.txt">Table of n, a(n) for n = 0..5000</a>
%F A385092 a(n) ~ Gamma(1/6) * exp(Pi*sqrt(n)) / (3 * 2^(10/3) * Pi^(5/6) * n^(7/12)).
%t A385092 nmax = 50; CoefficientList[Series[Sum[x^k*Product[(1+x^j)/(1-x^j), {j, 1, 6*k}], {k, 0, nmax}], {x, 0, nmax}], x]
%t A385092 nmax = 50; p = 1; q = 1; s = 1; Do[p = Expand[p*(1 - x^(6*k))*(1 - x^(6*k - 1))*(1 - x^(6*k - 2))*(1 - x^(6*k - 3))*(1 - x^(6*k - 4))*(1 - x^(6*k - 5))]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; q = Expand[q*(1 + x^(6*k))*(1 + x^(6*k - 1))*(1 + x^(6*k - 2))*(1 + x^(6*k - 3))*(1 + x^(6*k - 4))*(1 + x^(6*k - 5))]; 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 A385092 Cf. A035298, A385070.
%Y A385092 Cf. A207641, A385088, A385089, A385090, A385091.
%K A385092 nonn
%O A385092 0,3
%A A385092 _Vaclav Kotesovec_, Jun 17 2025