A324361 Total number of occurrences of n in the (signed) displacement sets of all permutations of [2n] divided by n!.

0, 1, 5, 49, 679, 12151, 266321, 6906257, 206788751, 7020426511, 266464077769, 11180868467209, 513915970996583, 25678820830238759, 1385874945753239969, 80341660921985676961, 4979071555472111291551, 328496221117149603559327, 22987138271050177264124441

Offset: 0

Author

Alois P. Heinz, Feb 23 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..366
• Wikipedia, Permutation

Crossrefs

Main diagonal of A324362.

Cf. A000142, A306234.

Programs

Maple

a:= proc(s) option remember; `if`(n<3, (3*n-1)*n/2,
      (8*n-12)*a(n-1)-(16*n^2-64*n+59)*a(n-2)-(4*n-10)*a(n-3))
    end:
seq(a(n), n=0..20);

Mathematica

A[n_, k_] := -Sum[(-1)^j*Binomial[n, j]*(n+k-j)!, {j, 1, n}]/k!;
a[n_] := A[n, n];
a /@ Range[0, 20] (* Jean-François Alcover, Oct 28 2021, after Alois P. Heinz in A324362 *)

Formula

a(n) = n! [x^n] (1-exp(-x))/(1-x)^(n+1).

a(n) = -1/n! * Sum_{j=1..n} (-1)^j * binomial(n,j) * (2n-j)!.

a(n) = (8*n-12)*a(n-1) - (16*n^2-64*n+59)*a(n-2) - (4*n-10)*a(n-3) for n > 2.

a(n) = A324362(n,n) = A306234(2n,n).