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A000240
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Rencontres numbers: permutations with exactly one fixed point.
(Formerly M2763 N1111)
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17
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1, 0, 3, 8, 45, 264, 1855, 14832, 133497, 1334960, 14684571, 176214840, 2290792933, 32071101048, 481066515735, 7697064251744, 130850092279665, 2355301661033952, 44750731559645107, 895014631192902120, 18795307255050944541, 413496759611120779880
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OFFSET
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1,3
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COMMENTS
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Contribution from Emeric Deutsch, Sep 06 2010: (Start)
a(n) is also the number of permutations of [n] having no circular succession. A circular succession in a permutation p of [n] is either a pair p(i), p(i+1), where p(i+1)=p(i)+1 or the pair p(n), p(1) if p(1)=p(n)+1. a(4)=8 because we have 1324, 1432, 4132, 2143, 2413, 3214, 3241, and 4321.
a(n)=A180188(n,0).
(End)
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REFERENCES
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S. K. Das and N. Deo, Rencontres graphs: a family of bipartite graphs, Fib. Quart., Vol. 25, No. 3, August 1987, 250-262.
I. Kaplansky, Symbolic solution of certain problems in permutations, Bull. Amer. Math. Soc., 50 (1944), 906-914.
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).
S. M. Tanny, Permutations and successions, J. Combinatorial Theory, Series A, 21 (1976), 196-202.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..100
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FORMULA
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E.g.f.: x*exp(-x)/(1-x). [corrected by Vaclav Kotesovec, Sep 26 2012]
a(n) = sum((-1)^k*n!/k!, k=0..n-1).
E.g.f.: x*A(x) where A(x) is the e.g.f. for A000166. - Geoffrey Critzer, Jan 14 2012
a(n) = n*a(n-1)-(-1)^n*n = A000166(n)-(-1)^n = n*A000166(n-1) = A000387(n+1)*2/(n+1) = A000449(n+2)*6/((n+1)*(n+2)).
a(n) = n*floor(((n-1)!+1)/e), n>1. - Gary Detlefs, Jul 13 2010
lim_{n->infinity} n!/a(n) = e = 2.71828...
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EXAMPLE
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a(3) = 3 because the permutations of (1,2,3) with one fixed point are (1,3,2), (3,2,1) and (2,1,3).
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MAPLE
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G(x):=exp(-x)/(1-x)*x: 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=1..22); # [From Zerinvary Lajos, Apr 03 2009]
A000240 := proc(n)
n!*add((-1)^k/k!, k=0..n-1) ;
end proc: # R. J. Mathar, Jul 09 2012
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MATHEMATICA
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Table[Subfactorial[n]*(n + 1), {n, 0, 21}] [From Zerinvary Lajos, Jul 10 2009]
Table[n!*Sum[(-1)^k/k!, {k, 0, n-1}], {n, 1, 25}] (* Vaclav Kotesovec, Sep 26 2012 *)
Table[n!*SeriesCoefficient[x*E^(-x)/(1-x), {x, 0, n}], {n, 1, 25}] (* Vaclav Kotesovec, Sep 26 2012 *)
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PROG
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(Python)
a=0
for i in range(1, 51):
. a = (a-(-1)**i)*i
. print a, # from Alex Ratushnyak, Apr 20 2012
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CROSSREFS
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Cf. A008290, A000166, A000387, etc.
A diagonal of A008291.
Cf. A180188.
Sequence in context: A074435 A039647 A071533 * A182390 A132103 A180508
Adjacent sequences: A000237 A000238 A000239 * A000241 A000242 A000243
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane, Simon Plouffe
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STATUS
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approved
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