A088026 Number of "sets of even lists" for even n, cf. A000262.

1, 2, 36, 1560, 122640, 15150240, 2695049280, 650948538240, 204637027795200, 81098021561356800, 39516616693678924800, 23204736106751520921600, 16152539421202464036556800, 13145716394493318293898240000, 12363004898960780220305909760000

Offset: 0

Author

Vladeta Jovovic, Nov 02 2003

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A052845, A088009, A206703.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1, add((i->
      b(n-i)*binomial(n-1, i-1)*i!)(2*j), j=1..n/2))
    end:
a:= n-> b(2*n):
seq(a(n), n=0..14);  # Alois P. Heinz, Feb 01 2022

Mathematica

Table[n!*SeriesCoefficient[E^(x^2/(1-x^2)),{x,0,n}],{n,0,40,2}] (* Vaclav Kotesovec, Feb 25 2014 *)

PARI

x='x+O('x^66); /* (half) that many terms */
v=Vec(serlaplace(exp(x^2/(1-x^2))));
vector(#v\2,n, v[2*n-1])
/* Joerg Arndt, Jul 29 2012 */

Formula

E.g.f.: exp(x^2/(1-x^2)) (even powers only, see PARI code).

E.g.f.: exp(x^2/(1-x^2)) = 4/(2-(x^2/(1-x^2))*G(0))-1 where G(k) = 1 - x^4/(x^4 + 4*(1-x^2)^2*(2*k+1)*(2*k+3)/G(k+1) ) (continued fraction). - Sergei N. Gladkovskii, Dec 10 2012

a(n) ~ 2^(2*n) * n^(2*n-1/4) * exp(sqrt(4*n)-2*n-1/2). - Vaclav Kotesovec, Feb 25 2014

D-finite with recurrence a(n) -2*(2*n-1)^2*a(n-1) +4*(n-1)*(n-2)*(2*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Feb 01 2022

a(n) = A206703(2n,n). - Alois P. Heinz, Feb 19 2022

Extensions

More terms from Joerg Arndt, Jul 29 2012.