A343353 Expansion of Product_{k>=1} 1 / (1 - x^k)^(8^(k-1)).

1, 1, 9, 73, 621, 5229, 44293, 374277, 3162447, 26694159, 225163687, 1897751079, 15983278059, 134519816427, 1131395821587, 9509592524371, 79880259426102, 670590654977718, 5626336598011078, 47179486350900358, 395410837699366686, 3312225325409475038, 27731588831310844302

Offset: 0

Author

Ilya Gutkovskiy, Apr 12 2021

Keywords

• nonn

Crossrefs

Cf. A034691, A104460, A144072, A343349, A343350, A343351, A343352, A343354, A343355.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      d*8^(d-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..22);  # Alois P. Heinz, Apr 12 2021

Mathematica

nmax = 22; CoefficientList[Series[Product[1/(1 - x^k)^(8^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 8^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 22}]

Formula

a(n) ~ exp(sqrt(n/2) - 1/16 + c/8) * 2^(3*n - 7/4) / (sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} 1/(j * (8^(j-1) - 1)). - Vaclav Kotesovec, Apr 12 2021