A343351 - OEIS

%I #10 Apr 12 2021 12:13:39

%S 1,1,7,43,280,1792,11586,74550,479892,3083640,19794678,126908502,

%T 812761299,5199586119,33230586285,212172173565,1353444677529,

%U 8626044781761,54931168743703,349524243121795,2222294161109422,14119034725444774,89639674321304392,568720801952770012

%N Expansion of Product_{k>=1} 1 / (1 - x^k)^(6^(k-1)).

%F a(n) ~ exp(sqrt(2*n/3) - 1/12 + c/6) * 6^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} 1/(j * (6^(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*6^(d-1), d=numtheory[divisors](j)), j=1..n)/n)

%p     end:

%p seq(a(n), n=0..23);  # _Alois P. Heinz_, Apr 12 2021

%t nmax = 23; CoefficientList[Series[Product[1/(1 - x^k)^(6^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]

%t a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 6^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 23}]

%Y Cf. A034691, A104460, A144070, A343349, A343350, A343352, A343353, A343354, A343355.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Apr 12 2021