A129153 Rencontres numbers: permutations with exactly 8 fixed points.

1, 0, 45, 330, 4455, 56628, 795795, 11930490, 190900710, 3245287760, 58415223438, 1109889169740, 22197783520770, 466153453732680, 10255375982438730, 235873647595600476, 5660967542295146895, 141524188557377590800

Offset: 8

Author

Zerinvary Lajos, May 25 2007

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 8..200
• Index entries for sequences related to permutations with fixed points

Crossrefs

Column k=8 of A008290.

Cf. A008291, A170942.

Programs

Maple

a:= n-> -sum((n-1)!*sum((-1)^k/(k-7)!, j=0..n-1), k=7..n-1)/8!: seq(a(n), n=8..30);

Mathematica

With[{nn=40}, Drop[CoefficientList[Series[Exp[-x]/(1 - x) x^8/8!, {x, 0, nn}], x]Range[0, nn]!, 8]] (* Vincenzo Librandi, Feb 19 2014 *)

PARI

x='x+O('x^66); Vec( serlaplace(exp(-x)/(1-x)*(x^8/8!)) ) \\ Joerg Arndt, Feb 19 2014

Formula

a(n) = A008290(n,8).

E.g.f.: exp(-x)/(1-x)*(x^8/8!). [Joerg Arndt, Feb 19 2014]

O.g.f.: (1/8!)*Sum_{k>=8} k!*x^k/(1 + x)^(k+1). - Ilya Gutkovskiy, Apr 15 2017

D-finite with recurrence (-n+8)*a(n) +n*(n-9)*a(n-1) +n*(n-1)*a(n-2)=0. - R. J. Mathar, Jul 06 2023

Extensions

Changed offset from 0 to 8 by Vincenzo Librandi, Feb 19 2014