A338463 Expansion of g.f.: (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^2.

1, 0, 2, 2, 3, 4, 5, 8, 9, 12, 15, 20, 23, 28, 36, 44, 52, 62, 76, 90, 106, 124, 149, 176, 203, 236, 279, 324, 372, 430, 499, 576, 657, 752, 867, 992, 1124, 1280, 1463, 1662, 1876, 2124, 2410, 2722, 3061, 3446, 3889, 4374, 4896, 5490, 6166, 6900, 7700, 8600

Offset: 2

Author

Ilya Gutkovskiy, Jan 31 2021

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 2..1002

Crossrefs

Cf. A000700, A022597, A047654, A048574, A073252, A327380, A338223.

Programs

Magma

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( (-1 + (&*[1+x^(2*j+1): j in [0..m+2]]) )^2 )); // G. C. Greubel, Sep 07 2023

Mathematica

nmax = 55; CoefficientList[Series[(-1 + Product[1/(1 + (-x)^k), {k, 1, nmax}])^2, {x, 0, nmax}], x] // Drop[#, 2] &
With[{k=2}, Drop[CoefficientList[Series[(2/QPochhammer[-1,-x] -1)^k, {x,0,80}], x], k]] (* G. C. Greubel, Sep 07 2023 *)

SageMath

m=80
def f(x): return (-1 + product(1+x^(2*j-1) for j in range(1,m+3)) )^2
def A338463_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
a=A338463_list(m); a[2:] # G. C. Greubel, Sep 07 2023

Formula

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

a(n) = Sum_{k=1..n-1} A000700(k) * A000700(n-k).

a(n) = A073252(n) - 2 * A000700(n) for n > 0.

a(n) = [x^n]( (2/QPochhammer(-1,-x) - 1)^2 ). - G. C. Greubel, Sep 07 2023