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%I #13 May 09 2021 02:29:32
%S 1,1,6,31,181,931,5431,27931,159806,834806,4697306,24478556,137931681,
%T 717306681,3989650431,20958791056,115494337931,604881056681,
%U 3333662306681,17439531447306,95396181837931,501716543166056,2725636758009806,14311071572462931,77793648720900431
%N Expansion of Product_{k>=1} 1 / (1 - 5^(k-1)*x^k).
%F a(n) = Sum_{k=0..n} p(n,k) * 5^(n-k), where p(n,k) = number of partitions of n into k parts.
%F a(n) ~ polylog(2, 1/5)^(1/4) * 5^(n - 1/2) * exp(2*sqrt(polylog(2, 1/5)*n)) / (sqrt(Pi)*n^(3/4)). - _Vaclav Kotesovec_, May 09 2021
%t nmax = 24; CoefficientList[Series[Product[1/(1 - 5^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%t Table[Sum[Length[IntegerPartitions[n, {k}]] 5^(n - k), {k, 0, n}], {n, 0, 24}]
%t a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 5^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 24}]
%Y Cf. A008284, A075900, A246937, A300579, A338673, A338675, A338676, A338677, A338678, A338679.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Apr 23 2021
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