A367977 Expansion of e.g.f. exp(-x) / (2 - exp(2*x)).

1, 1, 9, 73, 849, 12241, 211929, 4280473, 98806689, 2565862561, 74035143849, 2349822967273, 81361870604529, 3051889548205681, 123282485663042169, 5335770920836028473, 246332487897909570369, 12083010395805261921601, 627555570373369525058889, 34404109751876393769480073

Offset: 0

Author

Ilya Gutkovskiy, Dec 07 2023

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..380

Crossrefs

Cf. A000670, A052841, A080253, A216794, A344037, A367979, A367980, A367981, A367982, A367983.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 50);
Coefficients(R!(Laplace( Exp(-x)/(2-Exp(2*x)) ))) // G. C. Greubel, Jun 10 2024

Mathematica

nmax = 19; CoefficientList[Series[Exp[-x]/(2 - Exp[2 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (-1)^n + Sum[Binomial[n, k] 2^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]

SageMath

def A367977_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(-x)/(2-exp(2*x)) ).egf_to_ogf().list()
A367977_list(50) # G. C. Greubel, Jun 10 2024

Formula

a(n) = Sum_{k>=0} (2*k-1)^n / 2^(k+1).

a(n) = (-1)^n + Sum_{k=1..n} binomial(n,k) * 2^k * a(n-k).

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * 2^k * A000670(k).