A022694 Expansion of Product_{m>=1} (1 + m*q^m)^-2.

1, -2, -1, -2, 9, -2, 10, -16, 38, -98, 53, -116, 340, -434, 463, -990, 2378, -2792, 3660, -7058, 11454, -18900, 24104, -36206, 81623, -119400, 128194, -248062, 447066, -576154, 880401, -1415926, 2297516, -3724290, 4854450, -7299306, 13411402, -19129752, 25135890, -42841396, 71321016

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 2, g(n) = -n. - Seiichi Manyama, Dec 30 2017

Links

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

Crossrefs

Column k=2 of A297325.

Programs

Magma

Coefficients(&*[1/(1+m*x^m)^2:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 25 2018

Mathematica

With[{nmax=50}, CoefficientList[Series[Product[1/(1+k*q^k)^2, {k,1,nmax}], {q, 0, nmax}],q]] (* G. C. Greubel, Feb 22 2018 *)

PARI

apply(x->round(x), Vec(prodinf(m=1, 1/(1+m*q^m)^2+O(q^50)))) \\ Michel Marcus, Dec 30 2017

PARI

m=50; q='q+O('q^m); Vec(prod(n=1,m,1/(1+n*q^n)^2)) \\ G. C. Greubel, Feb 25 2018

Formula

G.f.: exp(-2*Sum_{j>=1} Sum_{k>=1} (-1)^(j+1)*k^j*x^(j*k)/j). - Ilya Gutkovskiy, Feb 08 2018