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Rencontres numbers: permutations with exactly 8 fixed points.
3

%I #23 Jul 06 2023 05:25:19

%S 1,0,45,330,4455,56628,795795,11930490,190900710,3245287760,

%T 58415223438,1109889169740,22197783520770,466153453732680,

%U 10255375982438730,235873647595600476,5660967542295146895,141524188557377590800

%N Rencontres numbers: permutations with exactly 8 fixed points.

%H Vincenzo Librandi, <a href="/A129153/b129153.txt">Table of n, a(n) for n = 8..200</a>

%H <a href="/index/Per#IntegerPermutationCatAuto">Index entries for sequences related to permutations with fixed points</a>

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

%F E.g.f.: exp(-x)/(1-x)*(x^8/8!). [_Joerg Arndt_, Feb 19 2014]

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

%F 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

%p 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);

%t 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 *)

%o (PARI) x='x+O('x^66); Vec( serlaplace(exp(-x)/(1-x)*(x^8/8!)) ) \\ _Joerg Arndt_, Feb 19 2014

%Y Column k=8 of A008290.

%Y Cf. A008291, A170942.

%K nonn

%O 8,3

%A _Zerinvary Lajos_, May 25 2007

%E Changed offset from 0 to 8 by _Vincenzo Librandi_, Feb 19 2014