login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A000475 Rencontres numbers: number of permutations of [n] with exactly 4 fixed points.
(Formerly M4969 N2132)
21
1, 0, 15, 70, 630, 5544, 55650, 611820, 7342335, 95449640, 1336295961, 20044438050, 320711010620, 5452087178160, 98137569209940, 1864613814984984, 37292276299704525, 783137802293789040, 17229031650463366195, 396267727960657413630 (list; graph; refs; listen; history; text; internal format)
OFFSET

4,3

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=4..100

FindStat - Combinatorial Statistic Finder, The number of fixed points of a permutation

FORMULA

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

MAPLE

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

MATHEMATICA

Table[Subfactorial[n - 4]*Binomial[n, 4], {n, 4, 23}] (* Zerinvary Lajos, Jul 10 2009 *)

PROG

(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

CROSSREFS

Cf. A008290, A000166, A000240, A000387, A000449, A129135.

A diagonal of A008291.

Cf. A170942.

Sequence in context: A124893 A126402 A053134 * A253476 A145053 A168298

Adjacent sequences:  A000472 A000473 A000474 * A000476 A000477 A000478

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

Formula corrected by Sean A. Irvine, Oct 26 2010

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified November 20 04:05 EST 2017. Contains 294959 sequences.