OFFSET
0,3
COMMENTS
The n-th term of the n-th inverse binomial transform of A000670.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..380
FORMULA
a(n) = n! * [x^n] exp(-n*x) / (2 - exp(x)).
a(n) = Sum_{k=0..n} binomial(n,k) * (-n)^(n - k) * A000670(k).
a(n) ~ (-1)^n * n^n / (2 - exp(-1)). - Vaclav Kotesovec, Dec 19 2019
MATHEMATICA
Table[Sum[(k - n)^n/2^(k + 1), {k, 0, Infinity}], {n, 0, 19}]
Table[HurwitzLerchPhi[1/2, -n, -n]/2, {n, 0, 19}]
Table[n! SeriesCoefficient[Exp[-n x]/(2 - Exp[x]), {x, 0, n}], {n, 0, 19}]
PROG
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 50);
A330603:= func< n | Coefficient(R!(Laplace( Exp(-n*x)/(2-Exp(x)) )), n) >;
[A330603(n): n in [0..30]]; // G. C. Greubel, Jun 12 2024
(SageMath) [factorial(n)*( exp(-n*x)/(2-exp(x)) ).series(x, n+1).list()[n] for n in (0..30)] # G. C. Greubel, Jun 12 2024
CROSSREFS
KEYWORD
sign
AUTHOR
Ilya Gutkovskiy, Dec 19 2019
STATUS
approved