A343355 Expansion of Product_{k>=1} 1 / (1 - x^k)^(10^(k-1)).

1, 1, 11, 111, 1166, 12166, 127436, 1332936, 13939651, 145683351, 1521743103, 15886781603, 165770328383, 1728861822083, 18022063489023, 187778810866043, 1955660195168328, 20358764860253028, 211849198103034998, 2203562708619192998, 22911457758236641451, 238129937419462634151

Offset: 0

Author

Ilya Gutkovskiy, Apr 12 2021

Keywords

• nonn

Comments

In general, if m > 1 and g.f. = Product_{k>=1} 1 / (1 - x^k)^(m^(k-1)), then a(n, m) ~ exp(2*sqrt(n/m) - 1/(2*m) + c(m)/m) * m^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c(m) = Sum_{j>=2} 1/(j * (m^(j-1) - 1)). - Vaclav Kotesovec, Apr 12 2021

Crossrefs

Cf. A034691, A104460, A292837, A343349, A343350, A343351, A343352, A343353, A343354.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      d*10^(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)^(10^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 10^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 21}]

Formula

a(n) ~ exp(sqrt(2*n/5) - 1/20 + c/10) * 10^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} 1/(j * (10^(j-1) - 1)). - Vaclav Kotesovec, Apr 12 2021