A283263 Expansion of exp( Sum_{n>=1} -sigma_3(n)*x^n/n ) in powers of x.

1, -1, -4, -5, -1, 21, 49, 81, 45, -121, -484, -997, -1344, -840, 1624, 6931, 15149, 23155, 23469, 2240, -57596, -168929, -322587, -461165, -450668, -64135, 985621, 2935044, 5718865, 8597971, 9683008, 5596899, -8414092, -37295629, -83336988, -141108721

Offset: 0

Author

Seiichi Manyama, Mar 04 2017

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Column k=2 of A283272.

Cf. A023871 (exp( Sum_{n>=1} sigma_3(n)*x^n/n )).

Cf. exp( Sum_{n>=1} -sigma_k(n)*x^n/n ): A010815 (k=1), A073592 (k=2), this sequence (k=3), A283264 (k=4), A283271 (k=5).

Programs

Mathematica

a[n_] := If[n<1, 1,-(1/n) * Sum[DivisorSigma[3, k] a[n - k], {k, n}]]; Table[a[n], {n, 0, 35}] (* Indranil Ghosh, Mar 16 2017 *)

PARI

a(n) = if(n<1, 1, -(1/n) * sum(k=1, n, sigma(k, 3) * a(n - k)));
for(n=0, 35, print1(a(n), ", ")) \\ Indranil Ghosh, Mar 16 2017

SageMath

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: -n^2)
print([b(n) for n in range(36)]) # Peter Luschny, Nov 11 2020

Formula

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

a(n) = -(1/n)*Sum_{k=1..n} sigma_3(k)*a(n-k).