A324353 Total number of occurrences of 3 in the (signed) displacement sets of all permutations of [n+3] divided by 3!.

0, 1, 7, 49, 375, 3181, 29843, 307833, 3468671, 42432445, 560365779, 7948580377, 120557659247, 1947336998829, 33378478735475, 605158251430681, 11571369420832383, 232739737871570173, 4912330587789969971, 108564708629365952505, 2507303342099915104559

Offset: 0

Author

Alois P. Heinz, Feb 23 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..447
• Wikipedia, Permutation

Crossrefs

Column k=3 of A324362.

Cf. A306234.

Programs

Maple

a:= n-> (k-> -add((-1)^j*binomial(n, j)*(n+k-j)!, j=1..n)/k!)(3):
seq(a(n), n=0..23);

Mathematica

m = 23;
CoefficientList[(1-Exp[-x])/(1-x)^4 + O[x]^(m+1), x]*Range[0, m]! (* Jean-François Alcover, May 03 2021 *)

Formula

E.g.f.: (1-exp(-x))/(1-x)^4.

a(n) = -1/3! * Sum_{j=1..n} (-1)^j * binomial(n,j) * (n+3-j)!.

a(n) = A306234(n+3,3).