A367981 Expansion of e.g.f. exp(-x) / (2 - exp(4*x)).

1, 3, 41, 699, 16145, 465843, 16129721, 651567339, 30080413985, 1562287110243, 90156154697801, 5722984840599579, 396313048731199025, 29731461242293159443, 2402032174787943537881, 207923994397338180409419, 19198138598327305954291265, 1883400035807194281432757443

Offset: 0

Author

Ilya Gutkovskiy, Dec 07 2023

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..345

Crossrefs

Cf. A000670, A052841, A285067, A328183, A367977, A367979, A367980, A367982, A367983.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!(Laplace( Exp(-x)/(2-Exp(4*x)) ))); // G. C. Greubel, Jun 11 2024

Mathematica

nmax = 17; CoefficientList[Series[Exp[-x]/(2 - Exp[4 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (-1)^n + Sum[Binomial[n, k] 4^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]

SageMath

def A367981_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(-x)/(2-exp(4*x)) ).egf_to_ogf().list()
A367981_list(40) # G. C. Greubel, Jun 11 2024

Formula

a(n) = Sum_{k>=0} (4*k-1)^n / 2^(k+1).

a(n) = (-1)^n + Sum_{k=1..n} binomial(n,k) * 4^k * a(n-k).

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * 4^k * A000670(k).