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A114492 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n and having k DDUU's, where U=(1,1), D=(1,-1) (0<=k<=floor(n/2)-1 for n>=2).

%I A114492 #29 Apr 11 2024 10:49:55
%S A114492 1,1,2,5,13,1,35,7,97,34,1,275,143,11,794,558,77,1,2327,2083,436,16,
%T A114492 6905,7559,2180,151,1,20705,26913,10051,1095,22,62642,94547,43796,
%U A114492 6758,268,1,190987,328943,183130,37402,2409,29,586219,1136218,742253,191408
%N A114492 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n and having k DDUU's, where U=(1,1), D=(1,-1) (0<=k<=floor(n/2)-1 for n>=2).
%C A114492 Rows 0 and 1 contain one term each; row n contains floor(n/2) terms (n>=2).
%C A114492 Row sums are the Catalan numbers (A000108). Column 0 yields A086581.
%C A114492 Sum(k*T(n,k),k=0..floor(n/2)-1) = binomial(2n-3,n-4) (A003516).
%H A114492 Alois P. Heinz, <a href="/A114492/b114492.txt">Rows n = 0..200, flattened</a>
%H A114492 Jean-Luc Baril, Pamela E. Harris, Kimberly J. Harry, Matt McClinton, and José L. Ramírez, <a href="https://arxiv.org/abs/2404.05672">Enumerating runs, valleys, and peaks in Catalan words</a>, arXiv:2404.05672 [math.CO], 2024. See p. 12.
%H A114492 FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000122">The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]].</a>
%H A114492 Toufik Mansour and Mark Shattuck, <a href="https://doi.org/10.26493/2590-9770.1552.b43">Counting occurrences of subword patterns in non-crossing partitions</a>, Art Disc. Appl. Math. (2022).
%H A114492 A. Sapounakis, I. Tasoulas and P. Tsikouras, <a href="http://dx.doi.org/10.1016/j.disc.2007.03.005">Counting strings in Dyck paths</a>, Discrete Math., 307 (2007), 2909-2924.
%F A114492 G.f.: G=G(t, z) satisfies z(t+z-tz)G^2-(1-2(1-t)z+(1-t)z^2)G+1-z+tz=0.
%e A114492 T(5,1) = 7 because we have UU(DDUU)DUDD, UU(DDUU)UDDD, UDUU(DDUU)DD, their mirror images and UUU(DDUU)DDD (the DDUU's are shown between parentheses).
%e A114492 Triangle starts:
%e A114492    1;
%e A114492    1;
%e A114492    2;
%e A114492    5;
%e A114492   13,  1;
%e A114492   35,  7;
%e A114492   97, 34, 1;
%e A114492   ...
%p A114492 G:=1/2/(-t*z-z^2+z^2*t)*(-1+2*z-2*t*z-z^2+z^2*t+sqrt(1+z^4-2*z^4*t+z^4*t^2-4*z+2*z^2-2*z^2*t)): Gser:=simplify(series(G,z=0,17)): P[0]:=1: for n from 1 to 14 do P[n]:=coeff(Gser,z^n) od: 1; 1; for n from 0 to 14 do seq(coeff(t*P[n],t^j),j=1..floor(n/2)) od; # yields sequence in triangular form
%t A114492 m = 15; G[_, _] = 0;
%t A114492 Do[G[t_, z_] = (-1 + z - t z - t z G[t, z]^2 - z^2 G[t, z]^2 + t z^2 G[t, z]^2)/(-1 + 2z - 2t z - z^2 + t z^2) + O[t]^Floor[m/2] + O[z]^m, {m}];
%t A114492 CoefficientList[#, t]& /@ Take[CoefficientList[G[t, z], z], m] // Flatten (* _Jean-François Alcover_, Oct 05 2019 *)
%Y A114492 Cf. A000108, A086581, A003516.
%K A114492 nonn,tabf
%O A114492 0,3
%A A114492 _Emeric Deutsch_, Dec 01 2005