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A243838 Number T(n,k) of Dyck paths of semilength n having exactly k (possibly overlapping) occurrences of the consecutive steps UDUUDDUUUUDUDDDDUDUD (with U=(1,1), D=(1,-1)); triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-1)/9)), read by rows. 2
1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16795, 1, 58783, 3, 208002, 10, 742865, 35, 2674314, 126, 9694383, 462, 35355954, 1716, 129638355, 6435, 477614390, 24310, 1767170813, 92376, 1, 6563767708, 352708, 4, 24464914958, 1352046, 16, 91477363405, 5200170, 65 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

UDUUDDUUUUDUDDDDUDUD is a Dyck path that contains all sixteen consecutive step patterns of length 4.

LINKS

Alois P. Heinz, Rows n = 0..350, flattened

EXAMPLE

Triangle T(n,k) begins:

:  0 :           1;

:  1 :           1;

:  2 :           2;

:  3 :           5;

:  4 :          14;

:  5 :          42;

:  6 :         132;

:  7 :         429;

:  8 :        1430;

:  9 :        4862;

: 10 :       16795,       1;

: 11 :       58783,       3;

: 12 :      208002,      10;

: 13 :      742865,      35;

: 14 :     2674314,     126;

: 15 :     9694383,     462;

: 16 :    35355954,    1716;

: 17 :   129638355,    6435;

: 18 :   477614390,   24310;

: 19 :  1767170813,   92376,  1;

: 20 :  6563767708,  352708,  4;

: 21 : 24464914958, 1352046, 16;

MAPLE

b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,

     `if`(x=0, 1, expand(b(x-1, y+1, [2, 2, 4, 5, 2, 4,

       8, 9, 10, 11, 2, 13, 5, 4, 2, 2, 18, 2, 20, 5][t])

      +`if`(t=20, z, 1) *b(x-1, y-1, [1, 3, 1, 3, 6, 7,

       1, 3, 3, 3, 12, 1, 14, 15, 16, 17, 1, 19, 1, 3][t]))))

    end:

T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)):

seq(T(n), n=0..30);

MATHEMATICA

b[x_, y_, t_] := b[x, y, t] = If[x == 0, 1, Expand[If[y >= x - 1, 0, b[x - 1, y + 1, {2, 2, 4, 5, 2, 4, 8, 9, 10, 11, 2, 13, 5, 4, 2, 2, 18, 2, 20, 5}[[t]]]] + If[t == 20, z, 1]*If[y == 0, 0, b[x - 1, y - 1, {1, 3, 1, 3, 6, 7, 1, 3, 3, 3, 12, 1, 14, 15, 16, 17, 1, 19, 1, 3}[[t]]]]]];

T[n_] := CoefficientList[b[2n, 0, 1], z];

T /@ Range[0, 30] // Flatten (* Jean-Fran├žois Alcover, Mar 27 2021, after Alois P. Heinz *)

CROSSREFS

Row sums give A000108.

T(736522,k) = A243752(736522,k).

T(n,0) = A243753(n,736522).

Cf. A243820.

Sequence in context: A261591 A291823 A287972 * A242450 A211216 A261592

Adjacent sequences:  A243835 A243836 A243837 * A243839 A243840 A243841

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Jun 11 2014

STATUS

approved

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Last modified August 12 00:34 EDT 2022. Contains 356067 sequences. (Running on oeis4.)