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A000449 Rencontres numbers: number of permutations of [n] with exactly 3 fixed points.
(Formerly M4700 N2009)
18
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((-1)^j*n!/(3!*j!), j=2..n-3).

For n >= 3 a(n) = C(n, 3) * A000166(n-3) = 1/6 * n! * sum((-1)^k /k!, k=0..n-3). - 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

a:=n->sum(n!*sum((-1)^k/(k-2)!, j=0..n), k=2..n): seq(a(n)/3!, n=2..21); # Zerinvary Lajos, May 25 2007

# 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. A008290, A000166, A000240, A000387, A000475, 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

N. J. A. Sloane.

STATUS

approved

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Last modified June 24 15:07 EDT 2017. Contains 288697 sequences.