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A114938
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Number of permutations of the multiset {1,1,2,2,....,n,n} with no two consecutive terms equal.
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15
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1, 0, 2, 30, 864, 39480, 2631600, 241133760, 29083420800, 4467125013120, 851371260364800, 197158144895712000, 54528028997584665600, 17752366094818747392000, 6720318485119046923315200, 2927066537906697348594432000, 1453437879238150456164433920000
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OFFSET
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0,3
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COMMENTS
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a(n) is also the number of (0,1)-matrices A=(a_ij) of size n X 2n such that each row has exactly two 1's and each column has exactly one 1 and with the restriction that no 1 stands on the line from a_11 to a_22. - Shanzhen Gao, Feb 24 2010
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997. Chapter 2, Sieve Methods, Example 2.2.3, page 68.
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LINKS
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Seiichi Manyama, Table of n, a(n) for n = 0..238 (terms 1..100 from Andrew Woods)
H. Eriksson, A. Martin, Enumeration of Carlitz multipermutations, arXiv:1702.04177 [math.CO], 2017.
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FORMULA
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a(n) = Sum_{k=0..n} ((binomial(n, k)*(-1)^(n-k)*(n+k)!)/2^k).
a(n) = (-1)^n * n! * A000806(n), n>0. - Vladeta Jovovic, Nov 19 2009
a(n) = n*(2*n-1)*a(n-1) + (n-1)*n*a(n-2). - Vaclav Kotesovec, Aug 07 2013
a(n) ~ 2^(n+1)*n^(2*n)*sqrt(Pi*n)/exp(2*n+1). - Vaclav Kotesovec, Aug 07 2013
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EXAMPLE
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a(2) = 2 because there are two permutations of {1,1,2,2} avoiding equal consecutive terms: 1212 and 2121.
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MATHEMATICA
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Table[Sum[Binomial[n, i](2n-i)!/2^(n-i) (-1)^i, {i, 0, n}], {n, 0, 20}] (* Geoffrey Critzer, Jan 02 2013, and adapted to the extension by Stefano Spezia, Nov 15 2018 *)
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PROG
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(PARI) vector(20, n, sum(k=0, n, binomial(n, k)*(-1)^(n-k)*(n+k)!/2^k)) \\ Michel Marcus, Aug 10 2015
(MAGMA) I:=[0, 2]; [n le 2 select I[n] else n*(2*n-1)*Self(n-1) + (n-1)*n*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Aug 10 2015
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CROSSREFS
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Cf. A114939 = preferred seating arrangements of n couples.
Cf. A007060 = arrangements of n couples with no adjacent spouses; A007060(n) = 2^n * A114938(n) (this sequence).
Cf. A193638.
Sequence in context: A013525 A270531 A229781 * A082653 A274389 A186292
Adjacent sequences: A114935 A114936 A114937 * A114939 A114940 A114941
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KEYWORD
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nonn
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AUTHOR
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Hugo Pfoertner, Jan 08 2006
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EXTENSIONS
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a(0)=1 prepended by Seiichi Manyama, Nov 15 2018
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STATUS
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approved
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