A283580 Expansion of exp( Sum_{n>=1} A283535(n)/n*x^n ) in powers of x.

1, 1, 65, 19748, 16799044, 30535636881, 101591759812967, 558649739234980142, 4722932373908389412037, 58154498193439779564557624, 1000058469893323150011227885608, 23226158305362748824532880463596385, 708825166389400019044145225134521489486

Offset: 0

Author

Seiichi Manyama, Mar 11 2017

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..152

Crossrefs

Cf. Product_{k>=1} 1/(1 - x^k)^(k^(m*k)): A000041 (m=0), A023880 (m=1), A283579 (m=2), this sequence (m=3).

Cf. A283536 (Product_{k>=1} (1 - x^k)^(k^(3*k))).

Programs

Mathematica

A[n_] :=  Sum[d^(3*d + 1), {d, Divisors[n]}]; a[n_] := If[n==0, 1, (1/n)*Sum[A[k]*a[n - k], {k, n}]];Table[a[n], {n, 0, 12}] (* Indranil Ghosh, Mar 11 2017 *)

PARI

A(n) = sumdiv(n, d, d^(3*d + 1));
a(n) = if(n==0, 1, (1/n)*sum(k=1, n, A(k)*a(n - k)));
for(n=0, 12, print1(a(n),", ")) \\ Indranil Ghosh, Mar 11 2017

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^(k^(3*k)).

a(n) = (1/n)*Sum_{k=1..n} A283535(k)*a(n-k) for n > 0.

a(n) ~ n^(3*n) * (1 + exp(-3)/n^3). - Vaclav Kotesovec, Mar 17 2017