|
%I #9 Apr 12 2021 12:13:14
%S 1,1,6,31,171,921,5031,27281,148101,801901,4336902,23415777,126254962,
%T 679805112,3655679442,19634501447,105334380517,564471596667,
%U 3021754455157,16160029793032,86339725851558,460874548444683,2457961986888773,13097958657023523,69740119667456018
%N Expansion of Product_{k>=1} 1 / (1 - x^k)^(5^(k-1)).
%F 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
%p a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
%p d*5^(d-1), d=numtheory[divisors](j)), j=1..n)/n)
%p end:
%p seq(a(n), n=0..24); # _Alois P. Heinz_, Apr 12 2021
%t nmax = 24; CoefficientList[Series[Product[1/(1 - x^k)^(5^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
%t 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}]
%Y Cf. A034691, A104460, A144069, A343349, A343351, A343352, A343353, A343354, A343355.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Apr 12 2021
|
