A308650 Number of permutations of [2n] with n components.

1, 1, 7, 58, 531, 5226, 54598, 601924, 6985987, 85328266, 1097775922, 14897635468, 213581648046, 3238686925956, 51972937713900, 882473430354888, 15839021166164451, 300037212548146890, 5986554523174314490, 125537613562829696828, 2760474045847159393466

Offset: 0

Author

Alois P. Heinz, Jun 13 2019

Keywords

• nonn

Examples

			a(2) = 7: 1|342, 1|423, 1|432, 21|43, 231|4, 312|4, 321|4.
a(3) = 58: 1|2|4563, 1|2|4635, 1|2|4653, 1|2|5364, ..., 4213|5|6, 4231|5|6, 4312|5|6, 4321|5|6.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A059438, A085771, A131577, A263484.

Programs

Mathematica

p[n_] := p[n] = n! - Sum[k!*p[n - k], {k, 1, n - 1}];
(* T is A059438 *)
T[n_, k_] /; n < k = 0;
T[n_, 1] := p[n];
T[n_, k_] /; n >= k := T[n, k] = Sum[T[n - j, k - 1]*p[j], {j, 1, n}];
a[n_] := T[2n, n];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Aug 31 2021 *)

Formula

a(n) = [x^(2n)] (1-1/Sum_{j=0..2n} j!*x^j)^n.

a(n) = A059438(2n,n) = A085771(2n,n) = A263484(2n,n).

a(n) is odd <=> n in { A131577 }.

a(n) ~ sqrt(2*Pi) * n^(n + 5/2) / exp(n-1). - Vaclav Kotesovec, Jun 25 2019