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A114938 Number of permutations of the multiset {1,1,2,2,....,n,n} with no two consecutive terms equal. 15

%I

%S 1,0,2,30,864,39480,2631600,241133760,29083420800,4467125013120,

%T 851371260364800,197158144895712000,54528028997584665600,

%U 17752366094818747392000,6720318485119046923315200,2927066537906697348594432000,1453437879238150456164433920000

%N Number of permutations of the multiset {1,1,2,2,....,n,n} with no two consecutive terms equal.

%C 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

%D R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997. Chapter 2, Sieve Methods, Example 2.2.3, page 68.

%H Seiichi Manyama, <a href="/A114938/b114938.txt">Table of n, a(n) for n = 0..238</a> (terms 1..100 from Andrew Woods)

%H H. Eriksson, A. Martin, <a href="https://arxiv.org/abs/1702.04177">Enumeration of Carlitz multipermutations</a>, arXiv:1702.04177 [math.CO], 2017.

%F a(n) = Sum_{k=0..n} ((binomial(n, k)*(-1)^(n-k)*(n+k)!)/2^k).

%F a(n) = (-1)^n * n! * A000806(n), n>0. - _Vladeta Jovovic_, Nov 19 2009

%F a(n) = n*(2*n-1)*a(n-1) + (n-1)*n*a(n-2). - _Vaclav Kotesovec_, Aug 07 2013

%F a(n) ~ 2^(n+1)*n^(2*n)*sqrt(Pi*n)/exp(2*n+1). - _Vaclav Kotesovec_, Aug 07 2013

%e a(2) = 2 because there are two permutations of {1,1,2,2} avoiding equal consecutive terms: 1212 and 2121.

%t 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 *)

%o (PARI) vector(20, n, sum(k=0, n, binomial(n, k)*(-1)^(n-k)*(n+k)!/2^k)) \\ _Michel Marcus_, Aug 10 2015

%o (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

%Y Cf. A114939 = preferred seating arrangements of n couples.

%Y Cf. A007060 = arrangements of n couples with no adjacent spouses; A007060(n) = 2^n * A114938(n) (this sequence).

%Y Cf. A193638.

%K nonn

%O 0,3

%A _Hugo Pfoertner_, Jan 08 2006

%E a(0)=1 prepended by _Seiichi Manyama_, Nov 15 2018

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Last modified April 22 04:29 EDT 2019. Contains 322329 sequences. (Running on oeis4.)