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A092107 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having exactly k UUU's (triple rises) where U=(1,1). Rows have 1,1,1,2,3,4,5,... entries, respectively.

%I A092107 #54 May 06 2025 11:50:50
%S A092107 1,1,2,4,1,9,4,1,21,15,5,1,51,50,24,6,1,127,161,98,35,7,1,323,504,378,
%T A092107 168,48,8,1,835,1554,1386,750,264,63,9,1,2188,4740,4920,3132,1335,390,
%U A092107 80,10,1,5798,14355,17028,12507,6237,2200,550,99,11,1,15511,43252,57816
%N A092107 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having exactly k UUU's (triple rises) where U=(1,1). Rows have 1,1,1,2,3,4,5,... entries, respectively.
%C A092107 Column 0 gives the Motzkin numbers (A001006), column 1 gives A014532. Row sums are the Catalan numbers (A000108).
%C A092107 Equal to A171380*B (without the zeros), B = A007318. - _Philippe Deléham_, Dec 10 2009
%H A092107 Alois P. Heinz, <a href="/A092107/b092107.txt">Rows n = 0..150, flattened</a>
%H A092107 Jean Luc Baril, Rigoberto Flórez, and José L. Ramirez, <a href="http://jl.baril.u-bourgogne.fr/narayana.pdf">Generalized Narayana arrays, restricted Dyck paths, and related bijections</a>, Univ. Bourgogne (France, 2025). See p. 14.
%H A092107 Michael Bukata, Ryan Kulwicki, Nicholas Lewandowski, Lara Pudwell, Jacob Roth, and Teresa Wheeland, <a href="https://arxiv.org/abs/1812.07112">Distributions of Statistics over Pattern-Avoiding Permutations</a>, arXiv:1812.07112 [math.CO], 2018.
%H A092107 FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000118">The number of occurrences of the contiguous pattern [.,[.,[.,.]]].</a>
%H A092107 Lara Pudwell, <a href="http://permutationpatterns.com/slides/Pudwell.pdf">On the distribution of peaks (and other statistics)</a>, 16th International Conference on Permutation Patterns, Dartmouth College, 2018.
%H A092107 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 A092107 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.
%H A092107 Aristidis Sapounakis, Panagiotis Tsikouras, Ioannis Tasoulas, and Kostas Manes, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i2p2">Strings of Length 3 in Grand-Dyck Paths and the Chung-Feller Property</a>, Electr. J. Combinatorics, 19 (2012), #P2.
%H A092107 Yidong Sun, <a href="http://dx.doi.org/10.1016/j.disc.2004.07.002">The statistic "number of udu's" in Dyck paths</a>, Discrete Math., 287 (2004), 177-186.
%F A092107 G.f.: G(t, z) satisfies z(t+z-tz)G^2 - (1-z+tz)G + 1 = 0.
%F A092107 Sum_{k=0..n} T(n,k)*x^k = A001405(n), A001006(n), A000108(n), A033321(n) for x = -1, 0, 1, 2 respectively. - _Philippe Deléham_, Dec 10 2009
%e A092107 T(5,2) = 5 because we have (U[UU)U]DUDDDD, (U[UU)U]DDUDDD, (U[UU)U]DDDUDD, (U[UU)U]DDDDUD and UD(U[UU)U]DDDD, where U=(1,1), D=(1,-1); the triple rises are shown between parentheses.
%e A092107 [1],[1],[2],[4, 1],[9, 4, 1],[21, 15, 5, 1],[51, 50, 24, 6, 1],[127, 161, 98, 35, 7, 1]
%e A092107 Triangle starts:
%e A092107      1;
%e A092107      1;
%e A092107      2;
%e A092107      4,    1;
%e A092107      9,    4,    1;
%e A092107     21,   15,    5,    1;
%e A092107     51,   50,   24,    6,    1;
%e A092107    127,  161,   98,   35,    7,    1;
%e A092107    323,  504,  378,  168,   48,    8,    1;
%e A092107    835, 1554, 1386,  750,  264,   63,    9,    1;
%e A092107   2188, 4740, 4920, 3132, 1335,  390,   80,   10,    1;
%e A092107   ...
%p A092107 b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
%p A092107       `if`(x=0, 1, expand(b(x-1, y-1, min(t+1,2))*
%p A092107       `if`(t=2, z, 1) +b(x-1, y+1, 0))))
%p A092107     end:
%p A092107 T:= n->(p->seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 0)):
%p A092107 seq(T(n), n=0..12);  # _Alois P. Heinz_, Mar 11 2014
%t A092107 b[x_, y_, t_] := b[x, y, t] = If[y>x || y<0, 0, If[x == 0, 1, Expand[b[x-1, y-1, Min[t+1, 2]]*If[t == 2, z, 1] + b[x-1, y+1, 0]]]]; T[n_] := Function[{p}, Table[ Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 0]]; Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Apr 29 2015, after _Alois P. Heinz_ *)
%Y A092107 Cf. A000108, A001006, A001405, A007318, A014532, A033321, A171380, A243752, A243753.
%K A092107 nonn,tabf
%O A092107 0,3
%A A092107 _Emeric Deutsch_, Mar 29 2004