A243881 Number T(n,k) of Dyck paths of semilength n having exactly k (possibly overlapping) occurrences of the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)); triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-1)/4)), read by rows.

1, 1, 2, 5, 14, 41, 1, 129, 3, 419, 10, 1395, 35, 4737, 124, 1, 16338, 454, 4, 57086, 1684, 16, 201642, 6305, 65, 718855, 23781, 263, 1, 2583149, 90209, 1077, 5, 9346594, 343809, 4419, 23, 34023934, 1315499, 18132, 105, 124519805, 5050144, 74368, 472, 1

Offset

0,3

Comments

UDUUUDDDUD is the only Dyck path of semilength 5 that contains all eight consecutive step patterns of length 3.

Links

Alois P. Heinz, Rows n = 0..300, flattened

Example

Triangle T(n,k) begins:
:  0 :        1;
:  1 :        1;
:  2 :        2;
:  3 :        5;
:  4 :       14;
:  5 :       41,       1;
:  6 :      129,       3;
:  7 :      419,      10;
:  8 :     1395,      35;
:  9 :     4737,     124,     1;
: 10 :    16338,     454,     4;
: 11 :    57086,    1684,    16;
: 12 :   201642,    6305,    65;
: 13 :   718855,   23781,   263,   1;
: 14 :  2583149,   90209,  1077,   5;
: 15 :  9346594,  343809,  4419,  23;
: 16 : 34023934, 1315499, 18132, 105;

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, 6, 2, 4, 2, 10, 2][t])+`if`(t=10,
      z, 1)*b(x-1, y-1, [1, 3, 1, 3, 3, 7, 8, 9, 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..20);

Mathematica

b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x==0, 1, Expand[b[x-1, y+1, {2, 2, 4, 5, 6, 2, 4, 2, 10, 2}[[t]]] + If[t==10, z, 1]*b[x-1, y-1, {1, 3, 1, 3, 3, 7, 8, 9, 1, 3}[[t]]]]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 1]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Mar 31 2015, after Alois P. Heinz *)

Crossrefs

Columns k=0-10 give: A243870, A243871, A243872, A243873, A243874, A243875, A243876, A243877, A243878, A243879, A243880.

Row sums give A000108.

T(738,k) = A243752(738,k).

T(n,0) = A243753(n,738).

Cf. A243882.

Sequence in context: A007463 A159308 A163189 * A358416 A225691 A116846

Adjacent sequences: A243878 A243879 A243880 * A243882 A243883 A243884

Keyword

nonn,tabf

Author

Alois P. Heinz, Jun 13 2014

Status

approved