OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..420
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
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}]
PROG
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!(Laplace( Exp(-2*x)/(2-Exp(x)) ))); // G. C. Greubel, Jun 11 2024
(SageMath)
def A344037_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( exp(-2*x)/(2-exp(x)) ).egf_to_ogf().list()
A344037_list(40) # G. C. Greubel, Jun 11 2024
CROSSREFS
KEYWORD
sign
AUTHOR
Ilya Gutkovskiy, Aug 01 2021
STATUS
approved