cp's OEIS Frontend

A385069 G.f.: Sum_{k>=0} x^k * Product_{j=1..5*k} (1 + x^j).

%I A385069 #11 Jun 17 2025 03:34:57
%S A385069 1,1,2,3,5,7,10,13,17,22,28,36,45,56,69,86,106,130,159,192,232,279,
%T A385069 333,397,472,557,657,773,905,1059,1236,1437,1669,1935,2236,2582,2976,
%U A385069 3422,3930,4507,5157,5894,6728,7664,8721,9911,11246,12746,14430,16312,18421,20780
%N A385069 G.f.: Sum_{k>=0} x^k * Product_{j=1..5*k} (1 + x^j).
%H A385069 Vaclav Kotesovec, <a href="/A385069/b385069.txt">Table of n, a(n) for n = 0..5000</a>
%F A385069 a(n) ~ Gamma(1/5) * 3^(3/20) * exp(Pi*sqrt(n/3)) / (5 * 2^(6/5) * Pi^(4/5) * n^(7/20)).
%t A385069 nmax = 60; CoefficientList[Series[Sum[x^k*Product[1 + x^j, {j, 1, 5*k}], {k, 0, nmax}], {x, 0, nmax}], x]
%t A385069 nmax = 60; p = 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]]]; s += p*x^k;, {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x]
%Y A385069 Cf. A000009, A192433, A385067, A385068, A385070.
%Y A385069 Cf. A035297.
%K A385069 nonn
%O A385069 0,3
%A A385069 _Vaclav Kotesovec_, Jun 16 2025