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%I #35 Jul 06 2023 05:28:00
%S 1,0,21,112,1134,11088,122430,1468368,19090071,267258992,4008887883,
%T 64142201760,1090417436108,19627513841376,372922762997772,
%U 7458455259939936,156627560458759005,3445806330092671776,79253545592131484497,1902085094211155585424
%N Number of permutations of [n] with exactly 5 fixed points.
%H Vincenzo Librandi, <a href="/A129135/b129135.txt">Table of n, a(n) for n = 5..200</a>
%H FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/St000022">The number of fixed points of a permutation</a>
%H <a href="/index/Per#IntegerPermutationCatAuto">Index entries for sequences related to permutations with fixed points</a>
%F a(n) = A008290(n,5).
%F E.g.f.: exp(-x)/(1-x)*(x^5/5!). - _Zerinvary Lajos_, Apr 03 2009
%F 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
%F 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
%F O.g.f.: (1/5!)*Sum_{k>=5} k!*x^k/(1 + x)^(k+1). - _Ilya Gutkovskiy_, Apr 15 2017
%p 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);
%p 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
%t 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 *)
%o (PARI) x='x+O('x^66); Vec(serlaplace(exp(-x)/(1-x)*(x^5/5!))) \\ _Joerg Arndt_, Feb 17 2014
%o (Python)
%o from sympy import binomial
%o A129135_list, m, x = [], 1, 0
%o for n in range(5,21):
%o ....x, m = x*n + m*binomial(n,5), -m
%o ....A129135_list.append(x) # _Chai Wah Wu_, Nov 01 2014
%Y Cf. A008290, A008291, A170942.
%K nonn
%O 5,3
%A _Zerinvary Lajos_, May 25 2007
%E Offset corrected by _Susanne Wienand_, Feb 17 2014
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