A338674 Expansion of Product_{k>=1} 1 / (1 - 5^(k-1)*x^k).

1, 1, 6, 31, 181, 931, 5431, 27931, 159806, 834806, 4697306, 24478556, 137931681, 717306681, 3989650431, 20958791056, 115494337931, 604881056681, 3333662306681, 17439531447306, 95396181837931, 501716543166056, 2725636758009806, 14311071572462931, 77793648720900431

Offset: 0

Author

Ilya Gutkovskiy, Apr 23 2021

Keywords

• nonn

Crossrefs

Cf. A008284, A075900, A246937, A300579, A338673, A338675, A338676, A338677, A338678, A338679.

Programs

Mathematica

nmax = 24; CoefficientList[Series[Product[1/(1 - 5^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[IntegerPartitions[n, {k}]] 5^(n - k), {k, 0, n}], {n, 0, 24}]
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}]

Formula

a(n) = Sum_{k=0..n} p(n,k) * 5^(n-k), where p(n,k) = number of partitions of n into k parts.

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