A022661 Expansion of Product_{m>=1} (1-m*q^m).

1, -1, -2, -1, -1, 5, 1, 13, 4, 0, 2, -8, -61, -31, 13, -156, 21, 11, 223, 92, 91, 426, 972, 165, 141, -1126, 440, 1294, -4684, -2755, -5748, -2414, -6679, 10511, -10048, -19369, 19635, 22629, 14027, 76969, -1990, 40193, -10678, 75795, 215767, -54322, -40882

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign

Comments

Is a(9) the only occurrence of 0 in this sequence? - Robert Israel, Jun 02 2015

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A006906, A022629, A022693, A266964.

Programs

Magma

Coefficients(&*[(1-m*x^m):m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 18 2018

Maple

P:= mul(1-m*q^m,m=1..100):
S:= series(P,q,101):
seq(coeff(S,q,j),j=0..100); # Robert Israel, Jun 02 2015
# second Maple program:
b:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, i*b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Sean A. Irvine (after Alois P. Heinz), May 19 2019

Mathematica

nmax = 40; CoefficientList[Series[Product[1 - k*x^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 15 2015 *)
nmax = 40; CoefficientList[Series[Exp[-Sum[PolyLog[-j, x^j]/j, {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 15 2015 *)
(* More efficient program: *) nmax = 50; poly = ConstantArray[0, nmax+1]; poly[[1]] = 1; poly[[2]] = -1; Do[Do[poly[[j+1]] -= k*poly[[j-k+1]], {j, nmax, k, -1}];, {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 07 2016 *)

PARI

m=50; q='q+O('q^m); Vec(prod(n=1,m,(1-n*q^n))) \\ G. C. Greubel, Feb 18 2018