A295382 Expansion of e.g.f. exp(-2*x/(1 - x))/(1 - x).

1, -1, -2, -2, 8, 88, 592, 3344, 14464, 2944, -1121536, -21603584, -317969408, -4202380288, -51322677248, -562045749248, -4751724347392, -3419742961664, 1260396818661376, 45221885372727296, 1218206507254153216, 29421299633821057024, 669044215287581769728, 14528992234596624498688

Offset: 0

Author

Ilya Gutkovskiy, Nov 21 2017

Keywords

• sign

Links

• G. C. Greubel, Table of n, a(n) for n = 0..449
• Eric Weisstein's World of Mathematics, Laguerre Polynomial
• Wikipedia, Laguerre polynomials
• Index entries for sequences related to Laguerre polynomials

Crossrefs

Column k=2 of A295381.

Cf. A009940, A087912, A160623, A160624, A277423.

Programs

Magma

[Factorial(n)*(&+[(-1)^k*Binomial(n,k)*2^k/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 06 2018

Maple

a:=series(exp(-2*x/(1-x))/(1-x),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 27 2019

Mathematica

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

PARI

my(x='x+O('x^30)); Vec(serlaplace(exp(-2*x/(1-x))/(1-x))) \\ G. C. Greubel, Feb 06 2018

Formula

E.g.f.: exp(-2*x/(1 - x))/(1 - x).

a(n) = n!*Laguerre(n,2).

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

a(n) = n!*A160623(n)/A160624(n).

a(n) = Sum_{k=0..n} (-2)^(n-k)*k!*binomial(n,k)^2. - Ridouane Oudra, Jul 08 2025