A129135 Number of permutations of [n] with exactly 5 fixed points.

1, 0, 21, 112, 1134, 11088, 122430, 1468368, 19090071, 267258992, 4008887883, 64142201760, 1090417436108, 19627513841376, 372922762997772, 7458455259939936, 156627560458759005, 3445806330092671776, 79253545592131484497, 1902085094211155585424

Offset

5,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 5..200

FindStat - Combinatorial Statistic Finder, The number of fixed points of a permutation

Index entries for sequences related to permutations with fixed points

Formula

a(n) = A008290(n,5).

E.g.f.: exp(-x)/(1-x)*(x^5/5!). - Zerinvary Lajos, Apr 03 2009

a(n) = n*a(n-1) - (-1^n)*binomial(n,5) with a(n) = 0 for n = 0,1,2,3,4. - Chai Wah Wu, Nov 01 2014

D-finite with recurrence (-n+5)*a(n) +n*(n-6)*a(n-1) +n*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 02 2015

O.g.f.: (1/5!)*Sum_{k>=5} k!*x^k/(1 + x)^(k+1). - Ilya Gutkovskiy, Apr 15 2017

Maple

a:=n->sum((n-1)!*sum((-1)^k/(k-4)!, j=0..n-1), k=4..n-1)/5!: seq(a(n), n=5..24);
x:='x'; G(x):=exp(-x)/(1-x)*(x^5/5!): f[0]:=G(x): for n from 1 to 26 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n], n=5..24); # Zerinvary Lajos, Apr 03 2009

Mathematica

With[{nn=30}, Drop[CoefficientList[Series[Exp[-x]/(1-x) x^5/5!, {x, 0, nn}], x]Range[0, nn]!, 5]] (* Harvey P. Dale, Jan 22 2013 *)

Prog

(PARI) x='x+O('x^66); Vec(serlaplace(exp(-x)/(1-x)*(x^5/5!))) \\ Joerg Arndt, Feb 17 2014
(Python)
from sympy import binomial
A129135_list, m, x = [], 1, 0
for n in range(5, 21):
....x, m = x*n + m*binomial(n, 5), -m
....A129135_list.append(x) # Chai Wah Wu, Nov 01 2014

Crossrefs

Cf. A008290, A008291, A170942.

Sequence in context: A157265 A275916 A355510 * A158091 A121628 A306532

Adjacent sequences: A129132 A129133 A129134 * A129136 A129137 A129138

Keyword

nonn

Author

Zerinvary Lajos, May 25 2007

Extensions

Offset corrected by Susanne Wienand, Feb 17 2014

Status

approved