OFFSET
0,3
COMMENTS
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
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
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
REFERENCES
R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997. Chapter 2, Sieve Methods, Example 2.2.3, page 68.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..238 (terms 1..100 from Andrew Woods)
H. Eriksson and A. Martin, Enumeration of Carlitz multipermutations, arXiv:1702.04177 [math.CO], 2017.
FORMULA
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
a(n) = n! * A278990(n). - Alexander Burstein, May 16 2020
From G. C. Greubel, Sep 26 2023: (Start)
a(n) = (-1)^n * (i/e)*sqrt(2/Pi) * n! * BesselK(n+1/2, -1).
a(n) = [n! * (1/x) * p_{n+1}(x)]|_{x= -1} (See A104548 for p_{n}(x)).
E.g.f.: sqrt(Pi/(2*x)) * exp(-(1+x)^2/(2*x)) * erfi((1+x)/sqrt(2*x)).
Sum_{n >= 0} a(n)*x^n/(n!)^2 = exp(sqrt(1-2*x))/sqrt(1-2*x).
Sum_{n >= 0} a(n)*x^n/(n!*(n+1)!) = ( 1 - exp(-1 + sqrt(1-2*x)) )/x. (End)
EXAMPLE
a(2) = 2 because there are two permutations of {1,1,2,2} avoiding equal consecutive terms: 1212 and 2121.
MATHEMATICA
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 *)
Table[Length[Select[Permutations[Join[Range[n], Range[n]]], !MatchQ[#, {___, x_, x_, ___}]&]], {n, 0, 5}] (* Gus Wiseman, Jul 04 2020 *)
PROG
(PARI) A114938(n)=sum(k=0, n, binomial(n, k)*(-1)^(n-k)*(n+k)!/2^k);
vector(20, n, A114938(n-1)) \\ Michel Marcus, Aug 10 2015
(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
(SageMath)
def A114938(n): return (-1)^n*sum(binomial(n, k)*factorial(n+k)//(-2)^k for k in range(n+1))
[A114938(n) for n in range(31)] # G. C. Greubel, Sep 26 2023
CROSSREFS
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. A278990 = number of loopless linear chord diagrams with n chords.
Cf. A000806 = Bessel polynomial y_n(-1).
The version for multisets with prescribed multiplicities is A335125.
The version for prime indices is A335452.
Anti-run compositions are counted by A003242.
Anti-run compositions are ranked by A333489.
Inseparable partitions are counted by A325535.
Inseparable partitions are ranked by A335448.
Separable partitions are counted by A325534.
Separable partitions are ranked by A335433.
Other sequences involving the multiset {1,1,2,2,...,n,n}: A001147, A007717, A020555, A094574, A316972.
Row n=2 of A322093.
KEYWORD
nonn
AUTHOR
Hugo Pfoertner, Jan 08 2006
EXTENSIONS
a(0)=1 prepended by Seiichi Manyama, Nov 15 2018
STATUS
approved