A129149 Permutations with exactly 7 fixed points.

1, 0, 36, 240, 2970, 34848, 454740, 6362928, 95450355, 1527194240, 25962321528, 467321755680, 8879113408308, 177582268088640, 3729227629977720, 82043007859339296, 1886989180765048965, 45287740338360829056

Offset: 7

Author

Zerinvary Lajos, May 25 2007

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 7..200
• P. Barry and A. Hennessy, Four-term Recurrences, Orthogonal Polynomials and Riordan Arrays, Journal of Integer Sequences, 2012, article 12.4.2. - From _N. J. A. Sloane_, Sep 21 2012
• Index entries for sequences related to permutations with fixed points

Crossrefs

Cf. A008290, A008291, A170942.

Programs

Maple

a:=n->sum(n!*sum((-1)^k/(k-6)!, j=0..n), k=6..n): seq(a(n)/7!, n=6..24);
restart: G(x):=exp(-x)/(1-x)*(x^7/7!): 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=7..24); # Zerinvary Lajos, Apr 03 2009

Mathematica

With[{nn=40}, Drop[CoefficientList[Series[Exp[-x]/(1 - x) x^7/7!, {x, 0, nn}], x]Range[0, nn]!, 7]] (* Vincenzo Librandi, Feb 19 2014 *)

PARI

x='x+O('x^66); Vec( serlaplace(exp(-x)/(1-x)*(x^7/7!)) ) \\ Joerg Arndt, Feb 19 2014

Formula

a(n) = A008290(n,7).

E.g.f.: exp(-x)/(1-x)*(x^7/7!). [Zerinvary Lajos, Apr 03 2009]

Conjecture: (-n+7)*a(n) +n*(n-8)*a(n-1) +n*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 02 2015

O.g.f.: (1/7!)*Sum_{k>=7} k!*x^k/(1 + x)^(k+1). - Ilya Gutkovskiy, Apr 15 2017

D-finite with recurrence (-n+7)*a(n) +n*(n-8)*a(n-1) +n*(n-1)*a(n-2)=0. - R. J. Mathar, Jul 06 2023

Extensions

Changed offset from 0 to 7 by Vincenzo Librandi, Feb 19 2014

Edited by Joerg Arndt, Feb 19 2014