A295384 a(n) = n!*Sum_{k=0..n} (-1)^k*binomial(2*n,n-k)*n^k/k!.

1, 1, 0, -15, -112, -135, 9504, 152425, 610560, -27692847, -765107200, -6289891839, 213472972800, 9380264146825, 129378550468608, -3294028613874375, -226623617585053696, -4707649131227927775, 83803818828756418560, 9446689798312021406353, 277055229100887244800000

Offset: 0

Author

Ilya Gutkovskiy, Nov 21 2017

Keywords

• sign

Links

• G. C. Greubel, Table of n, a(n) for n = 0..400
• Eric Weisstein's World of Mathematics, Laguerre Polynomial
• Wikipedia, Laguerre polynomials
• Index entries for sequences related to Laguerre polynomials

Crossrefs

Cf. A006902, A277373, A277423, A295385.

Programs

Magma

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

Maple

a := n -> pochhammer(n, n)*hypergeom([1 - n], [n], n):
seq(simplify(a(n)), n = 0..20); # Peter Luschny, Mar 23 2023

Mathematica

Table[n! SeriesCoefficient[Exp[-n x/(1 - x)]/(1 - x)^(n + 1), {x, 0, n}], {n, 0, 20}]
Table[n! LaguerreL[n, n, n], {n, 0, 20}]
Table[(-1)^n HypergeometricU[-n, n + 1, n], {n, 0, 20}]
Join[{1}, Table[n! Sum[(-1)^k Binomial[2 n, n - k] n^k/k!, {k, 0, n}], {n, 1, 20}]]

PARI

for(n=0,30, print1(n!*sum(k=0,n, (-1)^k*binomial(2*n,n-k)*n^k/k!), ", ")) \\ G. C. Greubel, Feb 06 2018

Formula

a(n) = n! * [x^n] exp(-n*x/(1 - x))/(1 - x)^(n+1).

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

a(n) = Pochhammer(n, n)*hypergeom([1 - n], [n], n). - Peter Luschny, Mar 23 2023