A330603 a(n) = Sum_{k>=0} (k - n)^n / 2^(k + 1).

1, 0, 3, -14, 155, -1834, 27867, -492246, 10068459, -232990178, 6025718963, -172182404734, 5387697769467, -183214963001082, 6728091949444491, -265348057242998822, 11185888456798395563, -501937946696294628946, 23886968118494957119011, -1201674025637823778926414

Offset: 0

Author

Ilya Gutkovskiy, Dec 19 2019

Keywords

• sign

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

Crossrefs

Cf. A000670, A052841, A290219, A292916.

Programs

Magma

R:=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

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}]

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

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