A295383 - OEIS

%I #20 Sep 08 2022 08:46:20

%S 1,-4,72,-2400,117600,-7620480,614718720,-59364264960,6678479808000,

%T -857813628672000,123868287980236800,-19863969090648883200,

%U 3502679882984419737600,-673592285189311488000000,140299650258002307072000000,-31464534897861317399347200000

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

%H G. C. Greubel, <a href="/A295383/b295383.txt">Table of n, a(n) for n = 0..300</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre Polynomial</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>

%H <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>

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

%F a(n) ~ (-1)^n * 16^n * (n-1)! / Pi. - _Vaclav Kotesovec_, Nov 21 2017

%F From _Peter Luschny_, Nov 21 2017: (Start)

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

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

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

%F a(n) = (-1)^n*binomial(2*n,n)^2*n!. - _Alois P. Heinz_, Oct 02 2021

%p a := n -> (-16)^n*GAMMA(n+1/2)^2/(Pi*GAMMA(n+1)):

%p seq(a(n), n=0..15); # _Peter Luschny_, Nov 21 2017

%t Table[(2 n)! SeriesCoefficient[(-x/(1 - x))^n /((1 - x) n!), {x, 0, 2 n}], {n, 0, 15}]

%t nmax = 15; CoefficientList[Series[2 EllipticK[-16 x]/Pi, {x, 0, nmax}], x] Range[0, nmax]!

%t Table[(-16)^n*Gamma[n + 1/2]^2/(Pi*Gamma[n + 1]), {n,0,50}] (* _G. C. Greubel_, Feb 06 2018 *)

%o (PARI) for(n=0,30, print1(round((-16)^n*gamma(n+1/2)^2/(Pi*gamma(n+1))), ", ")) \\ _G. C. Greubel_, Feb 06 2018

%o (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

%Y Central terms of triangles A021009 and A021010.

%Y Cf. A144084.

%K sign

%O 0,2

%A _Ilya Gutkovskiy_, Nov 21 2017