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

0,3

Links

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

Wikipedia, Permutation

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).

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 *)

Crossrefs

Main diagonal of A324362.

Cf. A000142, A306234.

Sequence in context: A297513 A228511 A116873 * A089914 A267220 A052142

Adjacent sequences: A324358 A324359 A324360 * A324362 A324363 A324364

Keyword

nonn

Author

Alois P. Heinz, Feb 23 2019

Status

approved