A343350 Expansion of Product_{k>=1} 1 / (1 - x^k)^(5^(k-1)).

1, 1, 6, 31, 171, 921, 5031, 27281, 148101, 801901, 4336902, 23415777, 126254962, 679805112, 3655679442, 19634501447, 105334380517, 564471596667, 3021754455157, 16160029793032, 86339725851558, 460874548444683, 2457961986888773, 13097958657023523, 69740119667456018

Offset: 0

Author

Ilya Gutkovskiy, Apr 12 2021

Keywords

• nonn

Crossrefs

Cf. A034691, A104460, A144069, A343349, A343351, A343352, A343353, A343354, A343355.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      d*5^(d-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..24);  # Alois P. Heinz, Apr 12 2021

Mathematica

nmax = 24; CoefficientList[Series[Product[1/(1 - x^k)^(5^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 5^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 24}]

Formula

a(n) ~ exp(2*sqrt(n/5) - 1/10 + c/5) * 5^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} 1/(j * (5^(j-1) - 1)). - Vaclav Kotesovec, Apr 12 2021