This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A295383 #21 Feb 16 2025 08:33:52
%S A295383 1,-4,72,-2400,117600,-7620480,614718720,-59364264960,6678479808000,
%T A295383 -857813628672000,123868287980236800,-19863969090648883200,
%U A295383 3502679882984419737600,-673592285189311488000000,140299650258002307072000000,-31464534897861317399347200000
%N A295383 a(n) = (2*n)! * [x^(2*n)] (-x/(1 - x))^n/((1 - x)*n!).
%H A295383 G. C. Greubel, <a href="/A295383/b295383.txt">Table of n, a(n) for n = 0..300</a>
%H A295383 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre Polynomial</a>
%H A295383 Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>
%H A295383 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A295383 E.g.f.: 2*K(-16*x)/Pi, where K() is the complete elliptic integral of the first kind.
%F A295383 a(n) ~ (-1)^n * 16^n * (n-1)! / Pi. - _Vaclav Kotesovec_, Nov 21 2017
%F A295383 From _Peter Luschny_, Nov 21 2017: (Start)
%F A295383 a(n) = (-16)^n*Gamma(n+1/2)^2/(Pi*Gamma(n+1)).
%F A295383 a(n) = (-16)^n*binomial(n-1/2,-1/2)*Gamma(n+1/2)/sqrt(Pi).
%F A295383 a(n) ~ (-exp(-1)*n*16)^n/sqrt(n*Pi/2). (End)
%F A295383 a(n) = (-1)^n*binomial(2*n,n)^2*n!. - _Alois P. Heinz_, Oct 02 2021
%p A295383 a := n -> (-16)^n*GAMMA(n+1/2)^2/(Pi*GAMMA(n+1)):
%p A295383 seq(a(n), n=0..15); # _Peter Luschny_, Nov 21 2017
%t A295383 Table[(2 n)! SeriesCoefficient[(-x/(1 - x))^n /((1 - x) n!), {x, 0, 2 n}], {n, 0, 15}]
%t A295383 nmax = 15; CoefficientList[Series[2 EllipticK[-16 x]/Pi, {x, 0, nmax}], x] Range[0, nmax]!
%t A295383 Table[(-16)^n*Gamma[n + 1/2]^2/(Pi*Gamma[n + 1]), {n,0,50}] (* _G. C. Greubel_, Feb 06 2018 *)
%o A295383 (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 A295383 (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 A295383 Central terms of triangles A021009 and A021010.
%Y A295383 Cf. A144084.
%K A295383 sign
%O A295383 0,2
%A A295383 _Ilya Gutkovskiy_, Nov 21 2017