A123922 Number of 2143-avoiding Dumont paths of the 2nd kind of length 2n.

1, 1, 2, 6, 21, 84, 360, 1650, 7865, 39039, 198744, 1039584, 5534928, 30046752, 165257136, 922280634, 5199131025, 29644168125, 170375955750, 988180543350, 5768664340725, 33927954699600, 200617471267200, 1193673954039840

Offset: 0

Author

R. J. Mathar, Nov 20 2006

Keywords

• easy
• nonn

Examples

			For n=2, there are 3 Dumont permutations of the 2nd kind of length 2n=4, namely {2143,3142,4132}.
Avoiding 2143, the cardinality of this set is reduced to a(2)=2.
		

Links

• A. Burstein, S. Elizalde and T. Mansour, Restricted Dumont Permutations, Dyck Paths and Noncrossing Partitions, arXiv:math/0610234 [math.CO], 2006.

Crossrefs

Cf. A001469, A047749.

Programs

Mathematica

b[n_] := If[EvenQ[n], Binomial[3n/2, n/2]/(n+1), Binomial[(3n-1)/2, (n+1)/2 ]/n];
a[n_] := b[n] b[n+1];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jul 27 2018 *)

PARI

A047749(n)={ my(m=floor(n/2)); if(n % 2, binomial(3*m+1,m+1)/(2*m+1), binomial(3*m,m)/(2*m+1)); }
a(n)={ A047749(n)*A047749(n+1); }

Formula

a(n) = A047749(n)*A047749(n+1).

Conjecture: 16*n*(n+2)*(n+1)^2*a(n) -108*n*(n+1)*(2*n-1)*a(n-1) -9*(3*n-5)*(3*n-1)*(3*n-4)*(3*n-2)*a(n-2)=0. - R. J. Mathar, Jan 25 2013