A295383 a(n) = (2*n)! * [x^(2*n)] (-x/(1 - x))^n/((1 - x)*n!).

1, -4, 72, -2400, 117600, -7620480, 614718720, -59364264960, 6678479808000, -857813628672000, 123868287980236800, -19863969090648883200, 3502679882984419737600, -673592285189311488000000, 140299650258002307072000000, -31464534897861317399347200000

Offset: 0

Author

Ilya Gutkovskiy, Nov 21 2017

Keywords

• sign

Links

• G. C. Greubel, Table of n, a(n) for n = 0..300
• Eric Weisstein's World of Mathematics, Laguerre Polynomial
• Wikipedia, Laguerre polynomials
• Index entries for sequences related to Laguerre polynomials

Crossrefs

Central terms of triangles A021009 and A021010.

Cf. A144084.

Programs

Magma

R:= RealField(); [Round((-16)^n*Gamma(n+1/2)^2/(Pi(R)*Gamma(n+1) )): n in [0..30]]; // G. C. Greubel, Feb 06 2018

Maple

a := n -> (-16)^n*GAMMA(n+1/2)^2/(Pi*GAMMA(n+1)):
seq(a(n), n=0..15); # Peter Luschny, Nov 21 2017

Mathematica

Table[(2 n)! SeriesCoefficient[(-x/(1 - x))^n /((1 - x) n!), {x, 0, 2 n}], {n, 0, 15}]
nmax = 15; CoefficientList[Series[2 EllipticK[-16 x]/Pi, {x, 0, nmax}], x] Range[0, nmax]!
Table[(-16)^n*Gamma[n + 1/2]^2/(Pi*Gamma[n + 1]), {n,0,50}] (* G. C. Greubel, Feb 06 2018 *)

PARI

for(n=0,30, print1(round((-16)^n*gamma(n+1/2)^2/(Pi*gamma(n+1))), ", ")) \\ G. C. Greubel, Feb 06 2018

Formula

E.g.f.: 2*K(-16*x)/Pi, where K() is the complete elliptic integral of the first kind.

a(n) ~ (-1)^n * 16^n * (n-1)! / Pi. - Vaclav Kotesovec, Nov 21 2017

From Peter Luschny, Nov 21 2017: (Start)

a(n) = (-16)^n*Gamma(n+1/2)^2/(Pi*Gamma(n+1)).

a(n) = (-16)^n*binomial(n-1/2,-1/2)*Gamma(n+1/2)/sqrt(Pi).

a(n) ~ (-exp(-1)*n*16)^n/sqrt(n*Pi/2). (End)

a(n) = (-1)^n*binomial(2*n,n)^2*n!. - Alois P. Heinz, Oct 02 2021