A307060 Expansion of 1/(2 - Product_{k>=1} 1/(1 + x^k)).

1, -1, 1, -2, 4, -7, 12, -21, 38, -68, 120, -212, 377, -670, 1188, -2107, 3740, -6638, 11778, -20898, 37084, -65808, 116775, -207212, 367696, -652478, 1157815, -2054524, 3645730, -6469316, 11479734, -20370656, 36147506, -64143372, 113821732, -201975429, 358403220, -635982680, 1128544452, -2002589998

Offset: 0

Author

Ilya Gutkovskiy, Mar 21 2019

Keywords

• sign

Comments

Invert transform of A081362.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A081362, A299208, A304969, A307058.

Cf. A307057, A307058, A307059, A307062, A307063.

Programs

Magma

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1/(2 - (&*[1-x^(2*j-1): j in [1..m+2]])) )); // G. C. Greubel, Jan 24 2024

Mathematica

nmax = 39; CoefficientList[Series[1/(2 - Product[1/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

SageMath

m=80;
def f(x): return 1/( 2 - product(1-x^(2*j-1) for j in range(1,m+3)) )
def A307060_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A307060_list(m) # G. C. Greubel, Jan 24 2024

Formula

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

a(0) = 1; a(n) = Sum_{k=1..n} A081362(k)*a(n-k).

From G. C. Greubel, Jan 24 2024: (Start)

G.f.: 1/(2 - QPochhammer(x)/QPochhammer(x^2)).

G.f.: 1/(2 - x^(1/24)*eta(x)/eta(x^2)), where eta(x) is the Dedekind eta function. (End)