A243998 Number T(n,k) of Dyck paths of semilength n with exactly k (possibly overlapping) occurrences of some of the consecutive patterns of Dyck paths of semilength 3; triangle T(n,k), n>=0, 0<=k<=max(0,n-2), read by rows.

1, 1, 2, 0, 5, 1, 11, 2, 1, 33, 7, 1, 4, 90, 30, 7, 1, 11, 245, 142, 24, 6, 1, 29, 680, 570, 121, 24, 5, 1, 81, 1884, 2176, 578, 112, 25, 5, 1, 220, 5265, 7935, 2649, 580, 116, 25, 5, 1, 608, 14747, 28022, 11827, 2825, 602, 124, 25, 5, 1

Offset

0,3

Comments

The consecutive patterns 101010, 101100, 110010, 110100, 111000 are counted. Here 1=Up=(1,1), 0=Down=(1,-1).

Links

Alois P. Heinz, Rows n = 0..142, flattened

Example

T(3,1) = 5: 101010, 101100, 110010, 110100, 111000.
T(4,0) = 1: 11001100.
T(4,2) = 2: 10101010, 10110010.
T(5,0) = 1: 1110011000.
T(6,3) = 7: 101010101100, 101010110010, 101100101010, 101100101100, 110010101010, 110010110010, 110101010100.
Triangle T(n,k) begins:
:  0 :   1;
:  1 :   1;
:  2 :   2;
:  3 :   0,    5;
:  4 :   1,   11,    2;
:  5 :   1,   33,    7,    1;
:  6 :   4,   90,   30,    7,   1;
:  7 :  11,  245,  142,   24,   6,   1;
:  8 :  29,  680,  570,  121,  24,   5,  1;
:  9 :  81, 1884, 2176,  578, 112,  25,  5, 1;
: 10 : 220, 5265, 7935, 2649, 580, 116, 25, 5, 1;

Maple

b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,
      `if`(x=0, 1, expand(add(b(x-1, y-1+2*j, irem(2*t+j, 32))*
      `if`(2*t+j in {42, 44, 50, 52, 56}, z, 1), j=0..1))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 0)):
seq(T(n), n=0..14);

Mathematica

b[x_, y_, t_] := b[x, y, t] = If[y < 0 || y > x, 0,
     If[x == 0, 1, Expand[Sum[b[x-1, y-1 + 2j, Mod[2t+j, 32]]*
     Switch[2t+j, 42|44|50|52|56, z, _, 1], {j, 0, 1}]]]];
T[n_] := CoefficientList[b[2n, 0, 0], z];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Apr 30 2022, after Alois P. Heinz *)

Crossrefs

Column k=0 gives A243986.

Row sums give A000108.

Cf. A014486, A063171, A243966.

Sequence in context: A370896 A307438 A198926 * A290395 A082974 A167635

Adjacent sequences: A243995 A243996 A243997 * A243999 A244000 A244001

Keyword

nonn,tabf

Author

Alois P. Heinz, Jun 17 2014

Status

approved