A322378 Triangle read by rows: T(n,k) is the number of nondecreasing Dyck prefixes (i.e., left factors of nondecreasing Dyck paths) of length n and final height k (0 <= k <= n).

1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 2, 0, 3, 0, 1, 0, 5, 0, 4, 0, 1, 5, 0, 9, 0, 5, 0, 1, 0, 13, 0, 14, 0, 6, 0, 1, 13, 0, 26, 0, 20, 0, 7, 0, 1, 0, 34, 0, 45, 0, 27, 0, 8, 0, 1, 34, 0, 73, 0, 71, 0, 35, 0, 9, 0, 1, 0, 89, 0, 137, 0, 105, 0, 44, 0, 10, 0, 1, 89, 0, 201, 0, 234, 0, 148, 0, 54, 0, 11, 0, 1, 0, 233, 0, 402, 0, 373, 0, 201, 0, 65, 0, 12, 0, 1, 233, 0, 546, 0, 733, 0, 564, 0, 265, 0, 77, 0, 13, 0, 1, 0, 610, 0, 1149, 0, 1245, 0, 818, 0, 341, 0, 90, 0, 14, 0, 1

Offset

0,8

Links

Table of n, a(n) for n=0..135.

R. Flórez and J. L. Ramírez, Some enumerations on non-decreasing Motzkin paths, Australasian Journal of Combinatorics, 72(1) (2018), 138-154.

Formula

Riordan array: ((1 - 2*x^2)/(1 - 3*x^2 + x^4), (x*(1-x^2))/(1 - 2*x^2)).

Example

Triangle begins:
   1;
   0,   1;
   1,   0,   1;
   0,   2,   0,   1;
   2,   0,   3,   0,   1;
   0,   5,   0,   4,   0,   1;
   5,   0,   9,   0,   5,   0,   1;
   0,  13,   0,  14,   0,   6,   0,   1;
  13,   0,  26,   0,  20,   0,   7,   0,   1;
   0,  34,   0,  45,   0,  27,   0,   8,   0,   1;
  34,   0,  73,   0,  71,   0,  35,   0,   9,   0,   1;
   0,  89,   0, 137,   0, 105,   0,  44,   0,  10,   0,   1;
  89,   0, 201,   0, 234,   0, 148,   0,  54,   0,  11,   0,   1;
   0, 233,   0, 402,   0, 373,   0, 201,   0,  65,   0,  12,   0,   1;
  ...

Crossrefs

Columns k=0, 1 give A001519. Column k=2 gives A061667.

Cf. A322329, A322325.

Sequence in context: A049803 A331843 A330635 * A053121 A113408 A242653

Adjacent sequences: A322375 A322376 A322377 * A322379 A322380 A322381

Keyword

nonn,tabl

Author

José Luis Ramírez Ramírez, Dec 05 2018

Status

approved