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A000475
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Rencontres numbers: number of permutations of [n] with exactly 4 fixed points.
(Formerly M4969 N2132)
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22
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1, 0, 15, 70, 630, 5544, 55650, 611820, 7342335, 95449640, 1336295961, 20044438050, 320711010620, 5452087178160, 98137569209940, 1864613814984984, 37292276299704525, 783137802293789040, 17229031650463366195, 396267727960657413630
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OFFSET
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4,3
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REFERENCES
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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).
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LINKS
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T. D. Noe, Table of n, a(n) for n=4..100
FindStat - Combinatorial Statistic Finder, The number of fixed points of a permutation
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FORMULA
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a(n) = sum((-1)^j*n!/(4!*j!), j=2..n-4).
a(n) = A000166(n)*binomial(n+4, 4). - Robert Goodhand (robert(AT)rgoodhand.fsnet.co.uk), Nov 08 2001
E.g.f.: (exp(-x)/(1-x))*(x^4/4!). In general, for k fixed points:(exp(-x)/(1-x)) * (x^k/k!). - Wenjin Woan, Nov 22 2008
a(n) ~ n! * exp(-1)/24, in general a(n) ~ n! * exp(-1)/k!. - Vaclav Kotesovec, Mar 16 2014
a(n) = n*a(n-1) + (-1^n)*binomial(n,4) with a(n) = 0 for n = 0,1,2,3. - Chai Wah Wu, Nov 01 2014
Conjecture: (-n+4)*a(n) +n*(n-5)*a(n-1) +n*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 02 2015
O.g.f.: (1/24)*Sum_{k>=4} k!*x^k/(1 + x)^(k+1). - Ilya Gutkovskiy, Apr 13 2017
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MAPLE
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a:=n->sum(n!*sum((-1)^k/(k-3)!, j=0..n), k=3..n): seq(-a(n)/4!, n=3..22); # Zerinvary Lajos, May 25 2007
G(x):=exp(-x)/(1-x)*(x^4/4!): 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=4..23); # Zerinvary Lajos, Apr 03 2009
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MATHEMATICA
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Table[Subfactorial[n - 4]*Binomial[n, 4], {n, 4, 23}] (* Zerinvary Lajos, Jul 10 2009 *)
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PROG
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(PARI) x='x+O('x^66); Vec( serlaplace(exp(-x)/(1-x)*(x^4/4!)) ) \\ Joerg Arndt, Feb 19 2014
(Python)
from sympy import binomial
A000475_list, m, x = [], 1, 0
for n in range(4, 100):
x, m = x*n + m*binomial(n, 4), -m
A000475_list.append(x) # Chai Wah Wu, Nov 01 2014
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CROSSREFS
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Cf. A008290, A000166, A000240, A000387, A000449, A129135.
A diagonal of A008291.
Cf. A170942.
Sequence in context: A053134 A320917 A343871 * A253476 A308596 A145053
Adjacent sequences: A000472 A000473 A000474 * A000476 A000477 A000478
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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Formula corrected by Sean A. Irvine, Oct 26 2010
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STATUS
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approved
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