A344037 Expansion of e.g.f.: exp(-2*x) / (2 - exp(x)).

1, -1, 3, -1, 27, 119, 1203, 11759, 136587, 1771559, 25562403, 405657119, 7022893947, 131714582999, 2660335750803, 57570797728079, 1328913670528107, 32592691757218439, 846383665814342403, 23200396829831840639, 669421949061096575067, 20281206249626017421879

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, A007047, A008619, A052841, A330603, A346208.

Programs

Magma

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

Mathematica

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

SageMath

def A344037_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(-2*x)/(2-exp(x)) ).egf_to_ogf().list()
A344037_list(40) # G. C. Greubel, Jun 11 2024

Formula

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

a(n) = Sum_{k=0..n} (-1)^k * Stirling2(n,k) * k! * A008619(k).

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

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

a(n) ~ n! / (8 * log(2)^(n+1)). - Vaclav Kotesovec, Aug 15 2021