A324355 Total number of occurrences of 5 in the (signed) displacement sets of all permutations of [n+5] divided by 5!.

0, 1, 11, 109, 1115, 12151, 142205, 1788361, 24118967, 347811859, 5345895929, 87298986325, 1510075068419, 27590646911023, 531082929791861, 10743610293871681, 227906995674679535, 5059315590877577131, 117308151182930808977, 2835988521500605314829

Offset: 0

Author

Alois P. Heinz, Feb 23 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..445
• Wikipedia, Permutation

Crossrefs

Column k=5 of A324362.

Cf. A306234.

Programs

Maple

a:= n-> (k-> -add((-1)^j*binomial(n, j)*(n+k-j)!, j=1..n)/k!)(5):
seq(a(n), n=0..23);

Mathematica

m = 23;
CoefficientList[(1-Exp[-x])/(1-x)^6 + O[x]^(m+1), x]*Range[0, m]! (* Jean-François Alcover, May 03 2021 *)

Formula

E.g.f.: (1-exp(-x))/(1-x)^6.

a(n) = -1/5! * Sum_{j=1..n} (-1)^j * binomial(n,j) * (n+5-j)!.

a(n) = A306234(n+5,5).