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%I #26 Jan 10 2017 16:49:07
%S 1,3,10,32,104,345,1166,4004,13936,49062,174420,625328,2258416,
%T 8209045,30008790,110255100,406923360,1507973610,5608843020,
%U 20931740640,78354322800,294127079610,1106939020044,4175827174152,15787544777504
%N Expansion of (1+x^2*C)*C^3, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
%C a(n)=number of Dyck (n+3)-paths whose third from last upstep initiates a long ascent, n>=1. A long ascent is one consisting of 2 or more upsteps. For example, a(1)=3 counts UDuUUDDD, UDuUDUDD, UDuUDDUD (third from last upstep in small type). - _David Callan_, Dec 08 2004
%C For n>0 a(n)=number of Dyck (n+3)-paths whose 5th and 6th steps are DU. For example, a(1)=3 counts UDUUduDD, UUDUduDD, UUUDduDD. - _David Scambler_, Feb 14 2011
%C Let X_n be the set of all noncrossing set partitions of an n-element set which either do not contain {n-1,n} as a block, or which do not contain the block {n} whenever 1 and n-1 are in the same block. a(n) is the cardinality of X_{n+2}. For example, a(1)=3 counts 1|2|3, 13|2, 123. - _Henri Mühle_, Jan 10 2017
%H M. Bruce, M. Dougherty, M. Hlavacek, R. Kudo, I. Nicolas, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i3p32">A Decomposition of Parking Functions by Undesired Spaces</a>, The Electronic Journal of Combinatorics 23(3), 2016.
%H H. Mühle, <a href="https://arxiv.org/abs/1701.02109">Two Posets of Noncrossing Partitions Coming From Undesired Parking Spaces</a>, arXiv:1701.02109 [math.CO], 2017.
%F For n>1, a(n) = 3*A000245(n) + A000344(n) = (5/(n+3) + 9/(n-1))*binomial(2n,n-2).
%F (n+3)*a(n) + 2*(-2*n-3)*a(n-1) + 2*(-n+1)*a(n-2) + 4*(2*n-5)*a(n-3) = 0. - _R. J. Mathar_, Aug 25 2013
%t {1, 3}~Join~Table[(5/(n + 3) + 9/(n - 1))*Binomial[2 n, n - 2], {n, 2, 24}] (* _Michael De Vlieger_, Jan 10 2017 *)
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Jun 06 2002
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