A367979 Expansion of e.g.f. exp(-x) / (2 - exp(3*x)).

1, 2, 22, 278, 4822, 104342, 2709622, 82092278, 2842418902, 110720079062, 4792059271222, 228144844817078, 11849163703935382, 666694458859845782, 40397145162583154422, 2622634244645856386678, 181615748103175019442262, 13362823095925278064444502, 1041037845089466806646007222

Offset: 0

Author

Ilya Gutkovskiy, Dec 07 2023

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..360

Crossrefs

Cf. A000670, A052841, A151919, A328182, A346208, A367977, A367980, A367981, A367982, A367983.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!(Laplace( Exp(-x)/(2-Exp(3*x)) ))); // G. C. Greubel, Jun 11 2024

Mathematica

nmax = 18; CoefficientList[Series[Exp[-x]/(2 - Exp[3 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (-1)^n + Sum[Binomial[n, k] 3^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]

SageMath

def A367979_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(-x)/(2-exp(3*x)) ).egf_to_ogf().list()
A367979_list(40) # G. C. Greubel, Jun 11 2024

Formula

a(n) = Sum_{k>=0} (3*k-1)^n / 2^(k+1).

a(n) = (-1)^n + Sum_{k=1..n} binomial(n,k) * 3^k * a(n-k).

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * 3^k * A000670(k).