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A000387
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Rencontres numbers: permutations with exactly two fixed points.
(Formerly M4138 N1716)
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16
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1, 0, 6, 20, 135, 924, 7420, 66744, 667485, 7342280, 88107426, 1145396460, 16035550531, 240533257860, 3848532125880, 65425046139824, 1177650830516985, 22375365779822544, 447507315596451070, 9397653627525472260, 206748379805560389951
(list; graph; refs; listen; history; internal format)
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OFFSET
| 2,3
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COMMENTS
| Also: odd permutations of length n with no fixed points. - Martin Wohlgemuth (mail(AT)matroid.com), May 31 2003
Also number of cycles of length 2 in all derangements of [n]. Example: a(4)=6 because in the derangements of [4], namely (1432), (1342), (13)(24), (1423), (12)(34), (1243), (1234), (1324), and (14)(23), we have altogether 6 cycles of length 2. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 31 2009]
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REFERENCES
| G. Gordon and E. McMahon, Moving faces to other places: facet derangements, Amer. Math. Monthly, 117 (2010), 865-88.
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
| T. D. Noe, Table of n, a(n) for n=2..100
M. Wohlgemuth Derangements revisited
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FORMULA
| a(n) = sum((-1)^j*n!/(2!*j!), j=2..n-2)
a(n) = A000166(n-2)*binomial(n, 2). - David Wasserman (wasserma(AT)spawar.navy.mil), Aug 13 2004
Contribution from Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 22 2009: (Start)
E.g.f.: G=z^2*exp(-z)/[2(1-z)].
(End)
A000387(n) = A145221(n) [From Mark van Hoeij (hoeij(AT)math.fsu.edu), Jul 06 2010]
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EXAMPLE
| a(4)=6 because we have 1243, 1432, 1324, 4231, 3214, and 2134. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 31 2009]
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MAPLE
| a:=n->sum(n!*sum((-1)^k/(k-1)!, j=0..n), k=1..n): seq(-a(n)/2!, n=1..21); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 18 2007
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MATHEMATICA
| Table[Subfactorial[n - 2]*Binomial[n, 2], {n, 2, 22}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 10 2009]
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CROSSREFS
| Cf. A000240, A000449, A000475, A003221.
A diagonal of A008291.
a(n)+A003221(n)=A000166(n).
Sequence in context: A074013 A114959 A000386 * A145221 A027148 A095854
Adjacent sequences: A000384 A000385 A000386 * A000388 A000389 A000390
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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