A367980 Expansion of e.g.f. exp(-2*x) / (2 - exp(3*x)).

1, 1, 19, 217, 3835, 82801, 2150659, 65156617, 2256029515, 87878584801, 3803459964499, 181078683329017, 9404687464288795, 529155742667806801, 32063235363798322339, 2081586179439325213417, 144148514796485770141675, 10606079719868369436964801, 826272285216863547170504179

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, A328182, A344037, A355218, A367977, A367979, A367981, A367982, A367983.

Programs

Magma

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

Mathematica

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

SageMath

def A367980_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(-2*x)/(2-exp(3*x)) ).egf_to_ogf().list()
A367980_list(40) # G. C. Greubel, Jun 11 2024

Formula

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

a(n) = (-2)^n + Sum_{k=1..n} binomial(n,k) * 3^k * a(n-k).

a(n) = Sum_{k=0..n} binomial(n,k) * (-2)^(n-k) * 3^k * A000670(k).