A307067 Expansion of 1/(2 - Product_{k>=2} (1 + x^k)).

1, 0, 1, 1, 2, 4, 6, 12, 19, 36, 60, 108, 187, 328, 576, 1005, 1765, 3084, 5408, 9461, 16575, 29017, 50812, 88977, 155792, 272813, 477684, 836466, 1464654, 2564685, 4490833, 7863610, 13769463, 24110774, 42218847, 73926591, 129448088, 226667986, 396903536, 694991728

Offset: 0

Author

Ilya Gutkovskiy, Mar 22 2019

Keywords

• nonn

Comments

Invert transform of A025147.

Links

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

Crossrefs

Cf. A025147, A055887, A299106, A304969, A307057, A307060, A317536.

Programs

Magma

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

Maple

a:=series(1/(2-mul((1+x^k),k=2..100)),x=0,40): seq(coeff(a,x,n),n=0..39); # Paolo P. Lava, Apr 03 2019

Mathematica

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

SageMath

m=80;
def f(x): return 1/( 2 - product(1+x^j for j in range(2, m+3)) )
def A307067_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A307067_list(m) # G. C. Greubel, Jan 24 2024

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A025147(k)*a(n-k).

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

G.f.: (1+x)/(2*(1+x) - QP(x^2)/QP(x)), where QP(x) = QPochhammer(x).

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