A022695 Expansion of Product_{m>=1} (1 + m*q^m)^-3.

1, -3, 0, -1, 18, -12, 11, -54, 84, -218, 243, -270, 1046, -1524, 1692, -3547, 7722, -11868, 15364, -29130, 52416, -83467, 125514, -190716, 380406, -628290, 808218, -1394734, 2585895, -3784566, 5678826, -9514614, 15635424, -25331990, 37563810, -57387042, 100038145, -156346224

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) = 3, g(n) = -n. - Seiichi Manyama, Dec 30 2017

Links

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

Crossrefs

Column k=3 of A297325.

Programs

Magma

Coefficients(&*[1/(1+m*x^m)^3: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)^3, {k,1,nmax}], {q, 0, nmax}],q]] (* G. C. Greubel, Feb 25 2018 *)

PARI

first(n) = Vec(prod(m=1, n, (1+m*x^m)^(-3)) + O(x^n)) \\ Iain Fox, Dec 30 2017

Formula

G.f.: exp(-3*Sum_{j>=1} Sum_{k>=1} (-1)^(j+1)*k^j*x^(j*k)/j). - Ilya Gutkovskiy, Feb 08 2018