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%I #27 Jul 06 2023 05:26:25
%S 1,0,36,240,2970,34848,454740,6362928,95450355,1527194240,25962321528,
%T 467321755680,8879113408308,177582268088640,3729227629977720,
%U 82043007859339296,1886989180765048965,45287740338360829056
%N Permutations with exactly 7 fixed points.
%H Vincenzo Librandi, <a href="/A129149/b129149.txt">Table of n, a(n) for n = 7..200</a>
%H P. Barry and A. Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Barry1/barry202.html">Four-term Recurrences, Orthogonal Polynomials and Riordan Arrays</a>, Journal of Integer Sequences, 2012, article 12.4.2. - From _N. J. A. Sloane_, Sep 21 2012
%H <a href="/index/Per#IntegerPermutationCatAuto">Index entries for sequences related to permutations with fixed points</a>
%F a(n) = A008290(n,7).
%F E.g.f.: exp(-x)/(1-x)*(x^7/7!). [_Zerinvary Lajos_, Apr 03 2009]
%F Conjecture: (-n+7)*a(n) +n*(n-8)*a(n-1) +n*(n-1)*a(n-2)=0. - _R. J. Mathar_, Nov 02 2015
%F O.g.f.: (1/7!)*Sum_{k>=7} k!*x^k/(1 + x)^(k+1). - _Ilya Gutkovskiy_, Apr 15 2017
%F 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
%p a:=n->sum(n!*sum((-1)^k/(k-6)!, j=0..n), k=6..n): seq(a(n)/7!, n=6..24);
%p 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
%t 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 *)
%o (PARI) x='x+O('x^66); Vec( serlaplace(exp(-x)/(1-x)*(x^7/7!)) ) \\ _Joerg Arndt_, Feb 19 2014
%Y Cf. A008290, A008291, A170942.
%K nonn
%O 7,3
%A _Zerinvary Lajos_, May 25 2007
%E Changed offset from 0 to 7 by _Vincenzo Librandi_, Feb 19 2014
%E Edited by _Joerg Arndt_, Feb 19 2014
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