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 A000449 Rencontres numbers: number of permutations of [n] with exactly 3 fixed points. (Formerly M4700 N2009) 19
 1, 0, 10, 40, 315, 2464, 22260, 222480, 2447445, 29369120, 381798846, 5345183480, 80177752655, 1282844041920, 21808348713320, 392550276838944, 7458455259940905, 149169105198816960, 3132551209175157490, 68916126601853463240 (list; graph; refs; listen; history; text; internal format)
 OFFSET 3,3 REFERENCES J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 65. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Table of n, a(n) for n = 3..100 FindStat - Combinatorial Statistic Finder, The number of fixed points of a permutation FORMULA a(n) = Sum_{j=2..n-3} (-1)^j*n!/(3!*j!). 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 E.g.f.: 1/(exp(x)*(1-x))*(x^3)/6. - Wenjin Woan, Nov 20 2008 From Paul Weisenhorn, May 30 2010: (Start) a(n) = binomial(n,3)*A000166(n-3) with 3 fixed-points; a(n) = binomial(n,k)*A000166(n-k) with k fixed-points. (End) E.g.f.: x^3*exp(-x)/(3!*(1-x)). - Geoffrey Critzer, Nov 03 2012 a(n) ~ n! * exp(-1)/6. - Vaclav Kotesovec, Mar 17 2014 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 O.g.f.: (1/6)*Sum_{k>=3} k!*x^k/(1 + x)^(k+1). - Ilya Gutkovskiy, Apr 13 2017 MAPLE # with k fixed-points: G:=exp(-z)*z^k/((1-z)*k!: Gser:=series(G, z, 21): for n from k to 20 do a(n)=n!*coeff(Gser, z, n): end do: # Paul Weisenhorn, May 30 2010 MATHEMATICA Table[Subfactorial[n - 3]*Binomial[n, 3], {n, 3, 22}] (* Zerinvary Lajos, Jul 10 2009 *) PROG (PARI) x='x+O('x^66); Vec( serlaplace(exp(-x)/(1-x)*(x^3/3!)) ) \\ Joerg Arndt, Feb 19 2014 (Python) from __future__ import division A000449_list, m, x = [], 1, 0 for n in range(3, 21): ....x, m = x*n + m*(n*(n-1)*(n-2)//6), -m ....A000449_list.append(x) # Chai Wah Wu, Sep 23 2014 CROSSREFS Cf. A000166, A000240, A000387, A000475, A008290, A129135. A diagonal of A008291. Cf. A170942. Sequence in context: A060580 A118266 A054885 * A027274 A253674 A016082 Adjacent sequences:  A000446 A000447 A000448 * A000450 A000451 A000452 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified January 22 14:02 EST 2020. Contains 331149 sequences. (Running on oeis4.)