A283579 Expansion of exp( Sum_{n>=1} A283533(n)/n*x^n ) in powers of x.

1, 1, 17, 746, 66418, 9843707, 2187941520, 680615139257, 282199700198462, 150389915598653924, 100155578743010743914, 81505577512720707466924, 79580089689432499741178617, 91814299713761739807846854872

Offset: 0

Author

Seiichi Manyama, Mar 11 2017

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..214

Crossrefs

Cf. Product_{k>=1} 1/(1 - x^k)^(k^(m*k)): A000041 (m=0), A023880 (m=1), this sequence (m=2), A283580 (m=3).

Cf. A283534 (Product_{k>=1} (1 - x^k)^(k^(2*k))).

Programs

Mathematica

A[n_] :=  Sum[d^(2*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, 13}] (* Indranil Ghosh, Mar 11 2017 *)

PARI

A(n) = sumdiv(n, d, d^(2*d + 1));
a(n) = if(n==0, 1, (1/n)*sum(k=1, n, A(k)*a(n - k)));
for(n=0, 11, print1(a(n),", ")) \\ Indranil Ghosh, Mar 11 2017

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^(k^(2*k)).

a(n) = (1/n)*Sum_{k=1..n} A283533(k)*a(n-k) for n > 0.

a(n) ~ n^(2*n) * (1 + exp(-2)/n^2). - Vaclav Kotesovec, Mar 17 2017