A324354 Total number of occurrences of 4 in the (signed) displacement sets of all permutations of [n+4] divided by 4!.

0, 1, 9, 76, 679, 6576, 69299, 792926, 9812079, 130741156, 1867777339, 28494131106, 462487232519, 7959671021576, 144813873037539, 2777366346993766, 56009230972732639, 1184896664408025036, 26240470547134420619, 607133649024919944266, 14649976322598313989879

Offset: 0

Author

Alois P. Heinz, Feb 23 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..446
• Wikipedia, Permutation

Crossrefs

Column k=4 of A324362.

Cf. A306234.

Programs

Maple

a:= n-> (k-> -add((-1)^j*binomial(n, j)*(n+k-j)!, j=1..n)/k!)(4):
seq(a(n), n=0..23);

Mathematica

m = 23;
CoefficientList[(1-Exp[-x])/(1-x)^5 + O[x]^(m+1), x]*Range[0, m]! (* Jean-François Alcover, May 03 2021 *)

Formula

E.g.f.: (1-exp(-x))/(1-x)^5.

a(n) = -1/4! * Sum_{j=1..n} (-1)^j * binomial(n,j) * (n+4-j)!.

a(n) = A306234(n+4,4).