A193215 Number of Dyck paths of semilength n having the property that the heights of the first and the last peaks coincide.

1, 2, 3, 6, 14, 38, 113, 356, 1164, 3906, 13364, 46426, 163294, 580316, 2080475, 7515038, 27324014, 99920756, 367264130, 1356043388, 5027345564, 18706888196, 69841532210, 261545298848, 982175296016, 3697785571820, 13954630170720, 52776659865348, 200006396351216, 759386612309146, 2888310863702017

Offset: 1

Author

Volodymyr Mazorchuk, Aug 26 2011

Keywords

• nonn
• easy

Comments

a(n+1) - a(n) = A000958(n) (this reduces to David Callan's comment on A000958(n) from Aug 23 2011).

The sequence gives the trace of the matrix describing the statistics of Dyck paths of semilength n with respect to the heights of the first and the last peaks, see the paper of Baur and Mazorchuk.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• K. Baur and V. Mazorchuk, Combinatorial analogues of ad-nilpotent ideals for untwisted affine Lie algebras, arXiv:1108.3659 [math.RA], 2011.

Programs

Maple

for n from 1 by 1 to 100 do 1+sum(binomial(2*n-2-2*k, n-1-k)-binomial(2*n-2-2*k, n-1-2*k), k = 1 .. n-1) end do

Mathematica

Table[1+Sum[Binomial[2*n-2-2*k, n-1-k]-Binomial[2*n-2-2*k, n-1-2*k],{k,1,n-1}],{n,1,20}] (* Vaclav Kotesovec, Mar 21 2014 *)

PARI

a(n)=1+sum(k=1,n-1,binomial(2*n-2-2*k, n-1-k)-binomial(2*n-2-2*k, n-1-2*k));

Formula

a(n) = 1 + Sum_{i=1..n-1} A000958(i).

Recurrence: 2*n*(5*n-11)*a(n) = 3*(15*n^2 - 53*n + 40)*a(n-1) - 3*(5*n^2 - 21*n + 20)*a(n-2) - 2*(2*n-5)*(5*n-6)*a(n-3). - Vaclav Kotesovec, Mar 21 2014

a(n) ~ 5*4^n/(27*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 21 2014