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A193215 Number of Dyck paths of semilength n having the property that the heights of the first and the last peaks coincide. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

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

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 *)

PROG

(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));

CROSSREFS

Sequence in context: A049339 A157100 A081293 * A007611 A098641 A188775

Adjacent sequences:  A193212 A193213 A193214 * A193216 A193217 A193218

KEYWORD

nonn,easy

AUTHOR

Volodymyr Mazorchuk, Aug 26 2011

STATUS

approved

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Last modified May 13 14:47 EDT 2021. Contains 343860 sequences. (Running on oeis4.)