A343355 - OEIS

%I #9 Apr 12 2021 12:18:31

%S 1,1,11,111,1166,12166,127436,1332936,13939651,145683351,1521743103,

%T 15886781603,165770328383,1728861822083,18022063489023,

%U 187778810866043,1955660195168328,20358764860253028,211849198103034998,2203562708619192998,22911457758236641451,238129937419462634151

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

%C 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

%F 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

%p a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(

%p       d*10^(d-1), d=numtheory[divisors](j)), j=1..n)/n)

%p     end:

%p seq(a(n), n=0..21);  # _Alois P. Heinz_, Apr 12 2021

%t nmax = 21; CoefficientList[Series[Product[1/(1 - x^k)^(10^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]

%t 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}]

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

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Apr 12 2021