A054479 Number of sets of cycle graphs of 2n nodes where the 2-colored edges alternate colors.

1, 0, 6, 120, 6300, 514080, 62785800, 10676746080, 2413521910800, 700039083744000, 253445583029839200, 112033456760809584000, 59382041886244720843200, 37175286835046004765120000, 27139206193305890195912400000, 22852066417535931447551359680000

Offset: 0

Author

Christian G. Bower, Mar 29 2000

Keywords

• nonn

Comments

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

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A001147, A001818, A053871, A006712.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1, add(
      b(n-2*j)*binomial(n-1, 2*j-1)*(2*j-1)!, j=2..n/2))
    end:
a:= n-> b(2*n):
seq(a(n), n=0..15);  # Alois P. Heinz, Mar 06 2023

Mathematica

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 *)

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

Formula

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

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

a(n) ~ 2^(2*n+1) * n^(2*n) / exp(2*n+1/2). - Vaclav Kotesovec, Mar 29 2014