A110468 - OEIS

%I #49 Aug 01 2024 18:51:40

%S 1,3,40,1260,72576,6652800,889574400,163459296000,39520825344000,

%T 12164510040883200,4644631106519040000,2154334728240414720000,

%U 1193170003333152768000000,777776389315596582912000000,589450799582646796969574400000,513927415886120176107847680000000

%N a(n) = (2*n + 1)!/(n + 1).

%C Convolution of (-1)^n*n! and n! with interpolated zeros suppressed.

%C Denominator of absolute value of coefficient of 1/(x+n^2) in the partial fraction decomposition of 1/(x+1)*1/(x+4)*..*1/(x+n^2). - Joris Roos (jorisr(AT)gmx.de), Aug 07 2009

%C With offset = 1: a(n) is the number of permutations of {1,2,...,2n} composed of two cycles of length n. - _Geoffrey Critzer_, Nov 11 2012

%H Alois P. Heinz, <a href="/A110468/b110468.txt">Table of n, a(n) for n = 0..200</a>

%F E.g.f.: log((1-x)*(1+x))/(-x).

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

%F a(n) = Sum_{k = 0..2*n} k!*(-1)^k*(2*n-k)!.

%F Sum_{n>=0} 1/a(n) = e/2. - _Franz Vrabec_, Jan 17 2008

%F (n+1)*a(n) + 2*(-n^2)*(2*n+1)*a(n-1) = 0. - _R. J. Mathar_, Nov 15 2012

%F a(n) = Product_{i=1..n} (n+1-i)*(n+1+i). - _Vaclav Kotesovec_, Oct 21 2014

%F a(n) = A145877(2*n+2, n+1). - _Alois P. Heinz_, Apr 21 2017

%F a(n) = A346085(2*n+2, n+1). - _Alois P. Heinz_, Jul 04 2021

%F Sum_{n>=0} (-1)^n/a(n) = (cos(1) + sin(1))/2 = (1/2) * A143623. - _Amiram Eldar_, Feb 08 2022

%F a(p-1) == 1 (mod p), p a prime. - _Peter Bala_, Jul 29 2024

%t Table[(2n)!/(2n^2),{n,1,20}] (* _Geoffrey Critzer_, Nov 11 2012 *)

%o (PARI) for(n=0,50, print1((2*n+1)!/(n+1), ", ")) \\ _G. C. Greubel_, Aug 28 2017

%Y Cf. A094310, A143623, A145877, A202768, A346085.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Jul 21 2005

%E Simpler definition from _Robert Israel_, Jul 20 2006