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A000449 Rencontres numbers: number of permutations of [n] with exactly 3 fixed points.
(Formerly M4700 N2009)
19

%I M4700 N2009

%S 1,0,10,40,315,2464,22260,222480,2447445,29369120,381798846,

%T 5345183480,80177752655,1282844041920,21808348713320,392550276838944,

%U 7458455259940905,149169105198816960,3132551209175157490,68916126601853463240

%N Rencontres numbers: number of permutations of [n] with exactly 3 fixed points.

%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 65.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A000449/b000449.txt">Table of n, a(n) for n = 3..100</a>

%H FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/St000022">The number of fixed points of a permutation</a>

%F a(n) = Sum_{j=2..n-3} (-1)^j*n!/(3!*j!).

%F For n >= 3 a(n) = C(n, 3) * A000166(n-3) = 1/6 * n! * Sum_{k=0..n-3} (-1)^k/k!. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 14 2001

%F E.g.f.: 1/(exp(x)*(1-x))*(x^3)/6. - _Wenjin Woan_, Nov 20 2008

%F From _Paul Weisenhorn_, May 30 2010: (Start)

%F a(n) = binomial(n,3)*A000166(n-3) with 3 fixed-points;

%F a(n) = binomial(n,k)*A000166(n-k) with k fixed-points.

%F (End)

%F E.g.f.: x^3*exp(-x)/(3!*(1-x)). - _Geoffrey Critzer_, Nov 03 2012

%F a(n) ~ n! * exp(-1)/6. - _Vaclav Kotesovec_, Mar 17 2014

%F a(n) = n*a(n-1) - (-1^n)*n*(n-1)*(n-2)/6, a(n) = 0 for n= 0, 1, 2. - _Chai Wah Wu_, Sep 23 2014

%F O.g.f.: (1/6)*Sum_{k>=3} k!*x^k/(1 + x)^(k+1). - _Ilya Gutkovskiy_, Apr 13 2017

%p # with k fixed-points:

%p G:=exp(-z)*z^k/((1-z)*k!: Gser:=series(G,z,21):

%p for n from k to 20 do a(n)=n!*coeff(Gser,z,n): end do: # _Paul Weisenhorn_, May 30 2010

%t Table[Subfactorial[n - 3]*Binomial[n, 3], {n, 3, 22}] (* _Zerinvary Lajos_, Jul 10 2009 *)

%o (PARI) x='x+O('x^66); Vec( serlaplace(exp(-x)/(1-x)*(x^3/3!)) ) \\ _Joerg Arndt_, Feb 19 2014

%o (Python)

%o from __future__ import division

%o A000449_list, m, x = [], 1, 0

%o for n in range(3,21):

%o ....x, m = x*n + m*(n*(n-1)*(n-2)//6), -m

%o ....A000449_list.append(x) # _Chai Wah Wu_, Sep 23 2014

%Y Cf. A000166, A000240, A000387, A000475, A008290, A129135.

%Y A diagonal of A008291.

%Y Cf. A170942.

%K nonn,easy

%O 3,3

%A _N. J. A. Sloane_

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Last modified May 30 08:04 EDT 2020. Contains 334712 sequences. (Running on oeis4.)