A346208 Expansion of e.g.f.: exp(-3*x) / (2 - exp(x)).

1, -2, 6, -14, 54, -62, 966, 4786, 71574, 875938, 12810726, 202739986, 3511712694, 65856494338, 1330170266886, 28785391689586, 664456856787414, 16296345814039138, 423191833100881446, 11600198414334789586, 334710974532291679734, 10140603124807778534338

Offset: 0

Author

Ilya Gutkovskiy, Aug 01 2021

Keywords

• sign

Links

• G. C. Greubel, Table of n, a(n) for n = 0..420

Crossrefs

Cf. A000670, A002620, A052841, A259533, A330603, A344037.

Programs

Magma

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

Mathematica

nmax = 21; CoefficientList[Series[Exp[-3 x]/(2 - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
Table[HurwitzLerchPhi[1/2, -n, -3]/2, {n, 0, 21}]
a[n_] := a[n] = (-3)^n + Sum[Binomial[n, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 21}]

SageMath

def A346208_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(-3*x)/(2-exp(x)) ).egf_to_ogf().list()
A346208_list(40) # G. C. Greubel, Jun 11 2024

Formula

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

a(n) = Sum_{k=0..n} (-1)^k * Stirling2(n,k) * k! * A002620(k+2).

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

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

a(n) ~ n! / (16 * log(2)^(n+1)). - Vaclav Kotesovec, Aug 15 2021