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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

FindStat - Combinatorial Statistic Finder, The number of occurrences of the contiguous pattern [.,[.,[.,.]]].

A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.

Aristidis Sapounakis, Panagiotis Tsikouras, Ioannis Tasoulas, 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: A174135 A182903 A169840 * 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 November 19 03:32 EST 2017. Contains 294912 sequences.