A054479 - OEIS

%I #32 Mar 06 2023 22:05:11

%S 1,0,6,120,6300,514080,62785800,10676746080,2413521910800,

%T 700039083744000,253445583029839200,112033456760809584000,

%U 59382041886244720843200,37175286835046004765120000,27139206193305890195912400000,22852066417535931447551359680000

%N Number of sets of cycle graphs of 2n nodes where the 2-colored edges alternate colors.

%C Also number of permutations in the symmetric group S_2n in which cycle lengths are even and greater than 2, cf. A130915. - _Vladeta Jovovic_, Aug 25 2007

%C a(n) is also the number of ordered pairs of disjoint perfect matchings in the complete graph on 2n vertices. The sequence A006712 is the number of ordered triples of perfect matchings. - _Matt Larson_, Jul 23 2016

%H Vincenzo Librandi, <a href="/A054479/b054479.txt">Table of n, a(n) for n = 0..200</a>

%F If b(2n)=a(n) then e.g.f. of b is 1/(sqrt(exp(x^2)*(1-x^2))).

%F a(n) = 4^n*(n-1)*gamma(n+1/2)^2*hypergeom([2-n],[3/2-n],-1/2)/(Pi*(n-1/2)). - _Mark van Hoeij_, May 13 2013

%F a(n) ~ 2^(2*n+1) * n^(2*n) / exp(2*n+1/2). - _Vaclav Kotesovec_, Mar 29 2014

%p b:= proc(n) option remember; `if`(n=0, 1, add(

%p       b(n-2*j)*binomial(n-1, 2*j-1)*(2*j-1)!, j=2..n/2))

%p     end:

%p a:= n-> b(2*n):

%p seq(a(n), n=0..15);  # _Alois P. Heinz_, Mar 06 2023

%t Table[(n-1)*(2*n)!^2 * HypergeometricPFQ[{2-n},{3/2-n},-1/2] / (4^n*(n-1/2)*(n!)^2), {n, 0, 20}] (* _Vaclav Kotesovec_, Mar 29 2014 after _Mark van Hoeij_ *)

%o (PARI) x='x+O('x^66); v=Vec(serlaplace(1/(sqrt(exp(x^2)*(1-x^2))))); vector(#v\2,n,v[2*n-1]) \\ _Joerg Arndt_, May 13 2013

%Y Cf. A001147, A001818, A053871, A006712.

%K nonn

%O 0,3

%A _Christian G. Bower_, Mar 29 2000