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) = A008290(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
D-finite with recurrence (-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
KEYWORD
nonn
AUTHOR
EXTENSIONS
Formula corrected by Sean A. Irvine, Oct 26 2010
STATUS
approved