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 A129135 Number of permutations of [n] with exactly 5 fixed points. 15
 1, 0, 21, 112, 1134, 11088, 122430, 1468368, 19090071, 267258992, 4008887883, 64142201760, 1090417436108, 19627513841376, 372922762997772, 7458455259939936, 156627560458759005, 3445806330092671776, 79253545592131484497, 1902085094211155585424 (list; graph; refs; listen; history; text; internal format)
 OFFSET 5,3 LINKS Vincenzo Librandi, Table of n, a(n) for n = 5..200 FindStat - Combinatorial Statistic Finder, The number of fixed points of a permutation FORMULA E.g.f.: exp(-x)/(1-x)*(x^5/5!). - Zerinvary Lajos, Apr 03 2009 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 Conjecture: (-n+5)*a(n) +n*(n-6)*a(n-1) +n*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 02 2015 O.g.f.: (1/5!)*Sum_{k>=5} k!*x^k/(1 + x)^(k+1). - Ilya Gutkovskiy, Apr 15 2017 MAPLE 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); 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 MATHEMATICA 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 *) PROG (PARI) x='x+O('x^66); Vec(serlaplace(exp(-x)/(1-x)*(x^5/5!))) \\ Joerg Arndt, Feb 17 2014 (Python) from sympy import binomial A129135_list, m, x = [], 1, 0 for n in range(5, 21): ....x, m = x*n + m*binomial(n, 5), -m ....A129135_list.append(x) # Chai Wah Wu, Nov 01 2014 CROSSREFS Cf. A008290, A000166, A000240, A000387, A000449, A000475, A008291, A170942. Sequence in context: A255285 A157265 A275916 * A158091 A121628 A306532 Adjacent sequences:  A129132 A129133 A129134 * A129136 A129137 A129138 KEYWORD nonn AUTHOR Zerinvary Lajos, May 25 2007 EXTENSIONS Offset corrected by Susanne Wienand, Feb 17 2014 STATUS approved

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Last modified May 30 09:21 EDT 2020. Contains 334717 sequences. (Running on oeis4.)