A306535 Number of permutations p of [2n] having no index i with |p(i)-i| = n.

1, 1, 9, 265, 14833, 1334961, 176214841, 32071101049, 7697064251745, 2355301661033953, 895014631192902121, 413496759611120779881, 228250211305338670494289, 148362637348470135821287825, 112162153835443422680893595673, 97581073836835777732377428235481

Offset: 0

Author

Alois P. Heinz, Feb 22 2019

Keywords

• nonn

Comments

Also 0th term of the 2n-th forward differences of n!.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..225
• Wikipedia, Permutation

Crossrefs

Cf. A306512, A306675.

Cf. A000142, A047920, A068106.

Programs

Maple

b:= proc(n, k) b(n, k):= `if`(k=0, n!, b(n+1, k-1) -b(n, k-1)) end:
a:= n-> b(0, 2*n):
seq(a(n), n=0..23);
seq(simplify(KummerU(-2*n, -2*n, -1)), n=0..15); # Peter Luschny, May 10 2022

Mathematica

b[n_, k_] := b[n, k] = If[k == 0, n!, b[n + 1, k - 1] - b[n, k - 1]];
a[n_] := b[0, 2n];
a /@ Range[0, 23] (* Jean-François Alcover, Apr 02 2021, after Alois P. Heinz *)

Formula

a(n) = A306512(2n,n).

a(n) = (2n)! - A306675(n).

a(n) = KummerU(-2*n, -2*n, -1). - Peter Luschny, May 10 2022