A307059 Expansion of 1/(2 - Product_{k>=1} (1 - x^k)).

1, -1, 0, 1, -1, 1, -1, 1, 0, -2, 4, -4, 1, 3, -5, 4, -3, 3, -1, -6, 13, -12, 2, 9, -13, 10, -6, 6, -4, -9, 28, -30, 5, 25, -28, 5, 9, 7, -27, 11, 32, -47, 2, 51, -27, -74, 128, -34, -131, 183, -78, -15, -37, 97, 89, -480, 649, -242, -498, 904, -663, 223, -140, 169, 488, -1818

Offset: 0

Author

Ilya Gutkovskiy, Mar 21 2019

Keywords

• sign

Comments

Invert transform of A010815.

Alternating row sums of Riordan triangle (1, 1 - Product_{j>=1} (1-x^j) ), See A341418(n, m) without column {1, repeat(0)} for m = 0 and n >= 0. - Wolfdieter Lang, Feb 17 2021

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500

Crossrefs

Cf. A010815, A055887, A299105, A341418.

Cf. A307057, A307058, A307060, A307062, A307063.

Programs

Magma

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1/(2 - (&*[1 - x^j: j in [1..m+2]])) )); // G. C. Greubel, Sep 08 2023

Mathematica

nmax=65; CoefficientList[Series[1/(2 - Product[(1 - x^k), {k, nmax}]), {x, 0, nmax}], x]

SageMath

from sage.modular.etaproducts import qexp_eta
m=80;
def f(x): return 1/(2 - qexp_eta(QQ[['q']], m+2).subs(q=x) )
def A307059_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A307059_list(m) # G. C. Greubel, Sep 08 2023

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A010815(k)*a(n-k).

G.f.: 1/(2 - QPochhammer(x)). - G. C. Greubel, Sep 08 2023