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

0,3

Links

Table of n, a(n) for n=0..23.

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

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

Example

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.

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 *)

Prog

(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); }

Crossrefs

Cf. A001469, A047749.

Sequence in context: A150223 A150224 A150225 * A350798 A326276 A099947

Adjacent sequences: A123919 A123920 A123921 * A123923 A123924 A123925

Keyword

easy,nonn

Author

R. J. Mathar, Nov 20 2006

Status

approved