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%I #24 Mar 29 2017 09:36:52
%S 1,2,3,6,14,38,113,356,1164,3906,13364,46426,163294,580316,2080475,
%T 7515038,27324014,99920756,367264130,1356043388,5027345564,
%U 18706888196,69841532210,261545298848,982175296016,3697785571820,13954630170720,52776659865348,200006396351216,759386612309146,2888310863702017
%N Number of Dyck paths of semilength n having the property that the heights of the first and the last peaks coincide.
%C a(n+1) - a(n) = A000958(n) (this reduces to David Callan's comment on A000958(n) from Aug 23 2011).
%C 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.
%H G. C. Greubel, <a href="/A193215/b193215.txt">Table of n, a(n) for n = 1..1000</a>
%H K. Baur and V. Mazorchuk, <a href="http://arxiv.org/abs/1108.3659">Combinatorial analogues of ad-nilpotent ideals for untwisted affine Lie algebras</a>, arXiv:1108.3659 [math.RA], 2011.
%F a(n) = 1 + Sum_{i=1..n-1} A000958(i).
%F 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
%F a(n) ~ 5*4^n/(27*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 21 2014
%p 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
%t 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 *)
%o (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));
%K nonn,easy
%O 1,2
%A _Volodymyr Mazorchuk_, Aug 26 2011
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