A322325 -id:A322325 - cp's OEIS frontend

A322329 Triangle read by rows: T(n,k) is the number of nondecreasing Motzkin prefixes (i.e., left factors of nondecreasing Motzkin paths) of length n and final height k (0 <= k <= n).

1, 1, 1, 2, 2, 1, 4, 5, 3, 1, 9, 12, 9, 4, 1, 21, 30, 25, 14, 5, 1, 49, 74, 69, 44, 20, 6, 1, 115, 182, 185, 133, 70, 27, 7, 1, 269, 444, 488, 386, 230, 104, 35, 8, 1, 630, 1078, 1266, 1090, 718, 369, 147, 44, 9, 1, 1474, 2605, 3245, 3006, 2161, 1232, 560, 200, 54, 10, 1

Offset: 0

José Luis Ramírez Ramírez, Dec 03 2018

			Triangle begins:
     1;
     1,    1;
     2,    2,    1;
     4,    5,    3,    1;
     9,   12,    9,    4,    1;
    21,   30,   25,   14,    5,    1;
    49,   74,   69,   44,   20,    6,   1;
   115,  182,  185,  133,   70,   27,   7,   1;
   269,  444,  488,  386,  230,  104,  35,   8,  1;
   630, 1078, 1266, 1090,  718,  369, 147,  44,  9,  1;
  1474, 2605, 3245, 3006, 2161, 1232, 560, 200, 54, 10, 1;
  ...

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

Column k=0 gives A322325.

Cf. A097862, A283595.

Riordan array: ((1 - x - 2*x^2 + x^3)/(1 - 2*x - 2*x^2 + 3x^3 - x^5),(x*(1-x)^2*(1+x))/(1 - 2*x - x^2 + 2*x^3 - x^4)).

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).

Original entry on oeis.org

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

José Luis Ramírez Ramírez, Dec 05 2018

			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;
  ...

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

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

Cf. A322329, A322325.

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

cp's OEIS Frontend

A322329 Triangle read by rows: T(n,k) is the number of nondecreasing Motzkin prefixes (i.e., left factors of nondecreasing Motzkin paths) of length n and final height k (0 <= k <= n).

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