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A110468
a(n) = (2*n + 1)!/(n + 1).
12
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
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
a(p-1) == 1 (mod p), p a prime. - Peter Bala, Jul 29 2024
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
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Jul 21 2005
EXTENSIONS
Simpler definition from Robert Israel, Jul 20 2006
STATUS
approved