A110468 a(n) = (2*n + 1)!/(n + 1).

1, 3, 40, 1260, 72576, 6652800, 889574400, 163459296000, 39520825344000, 12164510040883200, 4644631106519040000, 2154334728240414720000, 1193170003333152768000000, 777776389315596582912000000, 589450799582646796969574400000, 513927415886120176107847680000000

Offset

0,2

Comments

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

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

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

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

Formula

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

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

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

Sum_{n>=0} 1/a(n) = e/2. - Franz Vrabec, Jan 17 2008

(n+1)*a(n) + 2*(-n^2)*(2*n+1)*a(n-1) = 0. - R. J. Mathar, Nov 15 2012

a(n) = Product_{i=1..n} (n+1-i)*(n+1+i). - Vaclav Kotesovec, Oct 21 2014

a(n) = A145877(2*n+2, n+1). - Alois P. Heinz, Apr 21 2017

a(n) = A346085(2*n+2, n+1). - Alois P. Heinz, Jul 04 2021

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

Mathematica

Table[(2n)!/(2n^2), {n, 1, 20}] (* Geoffrey Critzer, Nov 11 2012 *)

Prog

(PARI) for(n=0, 50, print1((2*n+1)!/(n+1), ", ")) \\ G. C. Greubel, Aug 28 2017

Crossrefs

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

Sequence in context: A204515 A012250 A094330 * A327356 A295612 A350545

Adjacent sequences: A110465 A110466 A110467 * A110469 A110470 A110471

Keyword

easy,nonn

Author

Paul Barry, Jul 21 2005

Extensions

Simpler definition from Robert Israel, Jul 20 2006

Status

approved