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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. 4
 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, 168, 48, 8, 1, 835, 1554, 1386, 750, 264, 63, 9, 1, 2188, 4740, 4920, 3132, 1335, 390, 80, 10, 1, 5798, 14355, 17028, 12507, 6237, 2200, 550, 99, 11, 1, 15511, 43252, 57816 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Column 0 gives the Motzkin numbers (A001006), column 1 gives A014532. Row sums are the Catalan numbers (A000108). Equal to A171380*B (without the zeros), B = A007318. - Philippe Deléham, Dec 10 2009 LINKS Alois P. Heinz, Rows n = 0..150, flattened Michael Bukata, Ryan Kulwicki, Nicholas Lewandowski, Lara Pudwell, Jacob Roth, and Teresa Wheeland, Distributions of Statistics over Pattern-Avoiding Permutations, arXiv:1812.07112 [math.CO], 2018. FindStat - Combinatorial Statistic Finder, The number of occurrences of the contiguous pattern [.,[.,[.,.]]]. Lara Pudwell, On the distribution of peaks (and other statistics), 16th International Conference on Permutation Patterns, Dartmouth College, 2018. Toufik Mansour and Mark Shattuck, Counting occurrences of subword patterns in non-crossing partitions, Art Disc. Appl. Math. (2022). A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924. Aristidis Sapounakis, Panagiotis Tsikouras, Ioannis Tasoulas, and Kostas Manes, Strings of Length 3 in Grand-Dyck Paths and the Chung-Feller Property, Electr. J. Combinatorics, 19 (2012), #P2. Yidong Sun, The statistic "number of udu's" in Dyck paths, Discrete Math., 287 (2004), 177-186. FORMULA G.f.: G(t, z) satisfies z(t+z-tz)G^2 - (1-z+tz)G + 1 = 0. 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 EXAMPLE 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. [1],[1],[2],[4, 1],[9, 4, 1],[21, 15, 5, 1],[51, 50, 24, 6, 1],[127, 161, 98, 35, 7, 1] Triangle starts: 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, 168, 48, 8, 1; 835, 1554, 1386, 750, 264, 63, 9, 1; 2188, 4740, 4920, 3132, 1335, 390, 80, 10, 1; ... MAPLE b:= proc(x, y, t) option remember; `if`(y>x or 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)))) end: T:= n->(p->seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 0)): seq(T(n), n=0..12); # Alois P. Heinz, Mar 11 2014 MATHEMATICA 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 *) CROSSREFS Cf. A000108, A001006, A001405, A007318, A014532, A033321, A171380, A243752, A243753. Sequence in context: A182903 A169840 A321461 * A114489 A101974 A097607 Adjacent sequences: A092104 A092105 A092106 * A092108 A092109 A092110 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Mar 29 2004 STATUS approved

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Last modified July 17 08:16 EDT 2024. Contains 374360 sequences. (Running on oeis4.)