A245141 Number of endofunctions f on [2n] that are self-inverse on [n].

1, 3, 50, 1626, 83736, 6026120, 571350096, 67996818960, 9862902275456, 1700092943088768, 342087177215788800, 79115601821198404352, 20779757607847901690880, 6133520505473954148381696, 2017134796016735182500521984, 733523863838078950241395968000

Offset: 0

Author

Alois P. Heinz, Jul 21 2014

Keywords

• nonn

Comments

a(n) counts endofunctions f:{1,...,2n}-> {1,...,2n} with f(f(i))=i for all i in {1,...,n}.

Examples

			a(1) = 3: (1,1), (1,2), (2,1).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A000085, A245348.

Column k=2 of A246070.

Programs

Maple

g:= proc(n) g(n):= `if`(n<2, 1, g(n-1)+(n-1)*g(n-2)) end:
a:= n-> add(binomial(n, i)^2*i!*g(n-i)*(2*n)^(n-i), i=0..n):
seq(a(n), n=0..20);

Mathematica

Join[{1}, Table[n! * Sum[Binomial[n,k] * 2^k * n^k* Sum[1/((k - 2*j)!*2^j*j!), {j, 0, Floor[k/2]}], {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Dec 05 2021 *)

Formula

a(n) = Sum_{i=0..n} C(n,i)^2 * i! * A000085(n-i) * (2*n)^(n-i).

a(n) = A245348(2n,n).