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A385070 G.f.: Sum_{k>=0} x^k * Product_{j=1..6*k} (1 + x^j).

%I A385070 #12 Jun 17 2025 03:35:40
%S A385070 1,1,2,3,5,7,10,14,18,23,30,38,48,60,74,91,112,137,166,202,244,294,
%T A385070 352,420,500,592,700,824,968,1133,1323,1541,1791,2077,2403,2776,3198,
%U A385070 3679,4226,4845,5546,6340,7236,8246,9385,10667,12108,13728,15545,17581,19860,22409
%N A385070 G.f.: Sum_{k>=0} x^k * Product_{j=1..6*k} (1 + x^j).
%C A385070 In general, for m>=1, if g.f. = Sum_{k>=0} x^k * Product_{j=1..m*k} (1 + x^j), then a(n) ~ Gamma(1/m) * 3^((m-2)/(4*m)) * exp(Pi*sqrt(n/3)) / (m * 2^(1 + 1/m) * Pi^(1 - 1/m) * n^((m+2)/(4*m))). - _Vaclav Kotesovec_, Jun 17 2025
%H A385070 Vaclav Kotesovec, <a href="/A385070/b385070.txt">Table of n, a(n) for n = 0..5000</a>
%F A385070 a(n) ~ Gamma(1/6) * exp(Pi*sqrt(n/3)) / (2^(13/6) * 3^(5/6) * Pi^(5/6) * n^(1/3)).
%t A385070 nmax = 60; CoefficientList[Series[Sum[x^k*Product[1 + x^j, {j, 1, 6*k}], {k, 0, nmax}], {x, 0, nmax}], x]
%t A385070 nmax = 60; p = 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]]]; s += p*x^k;, {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x]
%Y A385070 Cf. A000009, A192433, A385067, A385068, A385069.
%Y A385070 Cf. A035298.
%K A385070 nonn
%O A385070 0,3
%A A385070 _Vaclav Kotesovec_, Jun 16 2025