A317364 Expansion of e.g.f. exp(2*x/(1 + x)).

1, 2, 0, -4, 16, -48, 64, 800, -12288, 127232, -1150976, 9266688, -58726400, 68777984, 7510646784, -207794409472, 4241007640576, -77359570944000, 1321952191971328, -21274345818161152, 313768799799607296, -3838962981483839488, 21775623343518515200, 859024717017756205056

Offset: 0

Author

Ilya Gutkovskiy, Jul 26 2018

Keywords

• sign

Comments

Inverse Lah transform of the powers of 2 (A000079).

Links

• Robert Israel, Table of n, a(n) for n = 0..451
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000079, A052897, A111884.

Cf. A001263, A008297, A086915.

Programs

Magma

[n eq 0 select 1 else (-1)^n*Factorial(n)*Evaluate(LaguerrePolynomial(n, -1), 2): n in [0..25]]; // G. C. Greubel, Feb 23 2021

Maple

a:= proc(n) option remember; add((-1)^(n-k)*
      n!/k!*binomial(n-1, k-1)*2^k, k=0..n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 26 2018

Mathematica

nmax = 23; CoefficientList[Series[Exp[2 x/(1 + x)], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Product[Exp[-2 (-x)^k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n-k) Binomial[n-1, k-1] 2^k n!/k!, {k, 0, n}], {n, 0, 23}]
Join[{1}, Table[2 (-1)^(n+1) n! Hypergeometric1F1[1-n, 2, 2], {n, 23}]]

PARI

a(n) = if (n==0, 1, (-1)^n*n!*pollaguerre(n, -1, 2)); \\ Michel Marcus, Feb 23 2021

Sage

[1 if n==0 else (-1)^n*factorial(n)*gen_laguerre(n, -1, 2) for n in (0..25)] # G. C. Greubel, Feb 23 2021

Formula

E.g.f.: Product_{k>=1} exp(-2*(-x)^k).

a(n) = 2*(-1)^(n+1) * n! * Hypergeometric1F1([1-n], [2], 2).

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

(n^2 + n)*a(n) + 2*n*a(n+1) + a(n+2) = 0. - Robert Israel, Aug 18 2019

From G. C. Greubel, Feb 23 2021: (Start)

a(n) = (-1)^n * n! * Laguerre(n, -1, 2) for n > 0 with a(0) = 1.

a(n) = Sum_{k=0..n} (-1)^(n-k) * A086915(n, k).

a(n) = (-1)^n * Sum_{k=0..n} 2^k * A008297(n, k).

a(n) = Sum_{k=0..n} (-1)^(n-k) * (n-k+1)! * A001263(n, k). (End)