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

%I A114938 #80 Sep 27 2023 06:07:15
%S A114938 1,0,2,30,864,39480,2631600,241133760,29083420800,4467125013120,
%T A114938 851371260364800,197158144895712000,54528028997584665600,
%U A114938 17752366094818747392000,6720318485119046923315200,2927066537906697348594432000,1453437879238150456164433920000
%N A114938 Number of permutations of the multiset {1,1,2,2,...,n,n} with no two consecutive terms equal.
%C A114938 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
%C A114938 a(n) is the number of permutations of the multiset {1,1,2,2,...,n,n} with no fixed points. - _Alexander Burstein_, May 16 2020
%C A114938 Also the number of 2-uniform ordered set partitions of {1...2n} containing no two successive vertices in the same block. - _Gus Wiseman_, Jul 04 2020
%D A114938 R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997. Chapter 2, Sieve Methods, Example 2.2.3, page 68.
%H A114938 Seiichi Manyama, <a href="/A114938/b114938.txt">Table of n, a(n) for n = 0..238</a> (terms 1..100 from Andrew Woods)
%H A114938 H. Eriksson and A. Martin, <a href="https://arxiv.org/abs/1702.04177">Enumeration of Carlitz multipermutations</a>, arXiv:1702.04177 [math.CO], 2017.
%F A114938 a(n) = Sum_{k=0..n} ((binomial(n, k)*(-1)^(n-k)*(n+k)!)/2^k).
%F A114938 a(n) = (-1)^n * n! * A000806(n), n>0. - _Vladeta Jovovic_, Nov 19 2009
%F A114938 a(n) = n*(2*n-1)*a(n-1) + (n-1)*n*a(n-2). - _Vaclav Kotesovec_, Aug 07 2013
%F A114938 a(n) ~ 2^(n+1)*n^(2*n)*sqrt(Pi*n)/exp(2*n+1). - _Vaclav Kotesovec_, Aug 07 2013
%F A114938 a(n) = n! * A278990(n). - _Alexander Burstein_, May 16 2020
%F A114938 From _G. C. Greubel_, Sep 26 2023: (Start)
%F A114938 a(n) = (-1)^n * (i/e)*sqrt(2/Pi) * n! * BesselK(n+1/2, -1).
%F A114938 a(n) = [n! * (1/x) * p_{n+1}(x)]|_{x= -1} (See A104548 for p_{n}(x)).
%F A114938 E.g.f.: sqrt(Pi/(2*x)) * exp(-(1+x)^2/(2*x)) * erfi((1+x)/sqrt(2*x)).
%F A114938 Sum_{n >= 0} a(n)*x^n/(n!)^2 = exp(sqrt(1-2*x))/sqrt(1-2*x).
%F A114938 Sum_{n >= 0} a(n)*x^n/(n!*(n+1)!) = ( 1 - exp(-1 + sqrt(1-2*x)) )/x. (End)
%e A114938 a(2) = 2 because there are two permutations of {1,1,2,2} avoiding equal consecutive terms: 1212 and 2121.
%t A114938 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 *)
%t A114938 Table[Length[Select[Permutations[Join[Range[n],Range[n]]],!MatchQ[#,{___,x_,x_,___}]&]],{n,0,5}] (* _Gus Wiseman_, Jul 04 2020 *)
%o A114938 (PARI) A114938(n)=sum(k=0, n, binomial(n, k)*(-1)^(n-k)*(n+k)!/2^k);
%o A114938 vector(20, n, A114938(n-1)) \\ _Michel Marcus_, Aug 10 2015
%o A114938 (Magma) [1] cat [n le 2 select 2*(n-1) else n*(2*n-1)*Self(n-1) + (n-1)*n*Self(n-2): n in [1..20]]; // _Vincenzo Librandi_, Aug 10 2015
%o A114938 (SageMath)
%o A114938 def A114938(n): return (-1)^n*sum(binomial(n,k)*factorial(n+k)//(-2)^k for k in range(n+1))
%o A114938 [A114938(n) for n in range(31)] # _G. C. Greubel_, Sep 26 2023
%Y A114938 Cf. A114939 = preferred seating arrangements of n couples.
%Y A114938 Cf. A007060 = arrangements of n couples with no adjacent spouses; A007060(n) = 2^n * A114938(n) (this sequence).
%Y A114938 Cf. A104548, A193638.
%Y A114938 Cf. A278990 = number of loopless linear chord diagrams with n chords.
%Y A114938 Cf. A000806 = Bessel polynomial y_n(-1).
%Y A114938 The version for multisets with prescribed multiplicities is A335125.
%Y A114938 The version for prime indices is A335452.
%Y A114938 Anti-run compositions are counted by A003242.
%Y A114938 Anti-run compositions are ranked by A333489.
%Y A114938 Inseparable partitions are counted by A325535.
%Y A114938 Inseparable partitions are ranked by A335448.
%Y A114938 Separable partitions are counted by A325534.
%Y A114938 Separable partitions are ranked by A335433.
%Y A114938 Cf. A007716, A128695, A292884, A335126, A335434, A335451.
%Y A114938 Other sequences involving the multiset {1,1,2,2,...,n,n}: A001147, A007717, A020555, A094574, A316972.
%Y A114938 Row n=2 of A322093.
%K A114938 nonn
%O A114938 0,3
%A A114938 _Hugo Pfoertner_, Jan 08 2006
%E A114938 a(0)=1 prepended by _Seiichi Manyama_, Nov 15 2018