A292916 a(n) = n! * [x^n] exp(n*x)/(2 - exp(x)).

1, 2, 11, 94, 1083, 15666, 272451, 5532206, 128409707, 3352959850, 97259891163, 3102552150006, 107936130271899, 4066743353318114, 164961642651034547, 7167348523420169278, 332081754670735087275, 16343667009638859878298, 851478575825591156040843, 46814697307371602567813126

Offset: 0

Author

Ilya Gutkovskiy, Sep 26 2017

Keywords

• nonn

Comments

The n-th term of the n-th binomial transform of A000670.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..380
• N. J. A. Sloane, Transforms

Crossrefs

Main diagonal of A292915.

Cf. A000670, A330603.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 50);
A292916:= func< n | Coefficient(R!(Laplace( Exp(n*x)/(2-Exp(x)) )), n) >;
[A292916(n): n in [0..30]]; // G. C. Greubel, Jun 12 2024

Maple

b:= proc(n, k) option remember; k^n +add(
       binomial(n, j)*b(j, k), j=0..n-1)
    end:
a:= n-> b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 27 2017

Mathematica

Table[n! SeriesCoefficient[Exp[n x]/(2 - Exp[x]), {x, 0, n}], {n, 0, 19}]
Table[HurwitzLerchPhi[1/2, -n, n]/2, {n, 0, 19}]

PARI

a000670(n) = sum(k=0, n, k!*stirling(n, k, 2));
a(n) = 2^n*a000670(n)-sum(k=0, n-1, 2^k*(n-1-k)^n); \\ Seiichi Manyama, Dec 25 2023

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) = A292915(n,n).

a(n) ~ n! * 2^(n-1) / (log(2))^(n+1). - Vaclav Kotesovec, Sep 27 2017

a(n) = 2^n*A000670(n) - Sum_{k=0..n-1} 2^k*(n-1-k)^n. - Seiichi Manyama, Dec 25 2023