A343354 Expansion of Product_{k>=1} 1 / (1 - x^k)^(9^(k-1)).

1, 1, 10, 91, 865, 8155, 77251, 730435, 6905560, 65233120, 615847378, 5810270782, 54784324495, 516250199827, 4862041512625, 45765734635702, 430560567351208, 4048630897384450, 38051334554031551, 357459295903931045, 3356488167698692226, 31503001136703776561

Offset: 0

Author

Ilya Gutkovskiy, Apr 12 2021

Keywords

• nonn

Crossrefs

Cf. A034691, A104460, A144073, A343349, A343350, A343351, A343352, A343353, A343355.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      d*9^(d-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..21);  # Alois P. Heinz, Apr 12 2021

Mathematica

nmax = 21; CoefficientList[Series[Product[1/(1 - x^k)^(9^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 9^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 21}]

Formula

a(n) ~ exp(2*sqrt(n/9) - 1/18 + c/9) * 9^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} 1/(j * (9^(j-1) - 1)). - Vaclav Kotesovec, Apr 12 2021