A283580 - OEIS

%I #20 Mar 17 2017 07:45:16

%S 1,1,65,19748,16799044,30535636881,101591759812967,558649739234980142,

%T 4722932373908389412037,58154498193439779564557624,

%U 1000058469893323150011227885608,23226158305362748824532880463596385,708825166389400019044145225134521489486

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

%H Seiichi Manyama, <a href="/A283580/b283580.txt">Table of n, a(n) for n = 0..152</a>

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

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

%F a(n) ~ n^(3*n) * (1 + exp(-3)/n^3). - _Vaclav Kotesovec_, Mar 17 2017

%t 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 *)

%o (PARI) A(n) = sumdiv(n, d, d^(3*d + 1));

%o a(n) = if(n==0, 1, (1/n)*sum(k=1, n, A(k)*a(n - k)));

%o for(n=0, 12, print1(a(n),", ")) \\ _Indranil Ghosh_, Mar 11 2017

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

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

%K nonn

%O 0,3

%A _Seiichi Manyama_, Mar 11 2017