A322329 -id:A322329 - cp's OEIS frontend

A322325 Number of nondecreasing Motzkin paths of length n.

1, 1, 2, 4, 9, 21, 49, 115, 269, 630, 1474, 3450, 8073, 18893, 44212, 103465, 242125, 566617, 1325982, 3103035, 7261648, 16993545, 39767898, 93063924, 217786044, 509657890, 1192689641, 2791104828, 6531679192, 15285285161, 35770272112, 83708766611, 195893326791

Offset: 0

José Luis Ramírez Ramírez, Dec 03 2018

			For n=6 we have 49 paths. Among the A001006(6) = 51 Motzkin paths, the following two paths are not nondecreasing Motzkin paths: UHUDDH and UUDHDH.

R. Flórez and J. L. Ramírez, Some enumerations on non-decreasing Motzkin paths, Australasian Journal of Combinatorics, 72(1) (2018), 138-154.
Index entries for linear recurrences with constant coefficients, signature (2,2,-3,0,1)

Column k=0 of A322329.

Cf. A001006, A001519.

LinearRecurrence[{2, 2, -3, 0, 1}, {1, 1, 2, 4, 9}, 40] (* Amiram Eldar, Dec 03 2018 *)

a(n) = 2*a(n-1) + 2*a(n-2) - 3*a(n-3) + a(n-5), a(0)=1, a(1)=1, a(2)=2, a(3)=4, a(4)=9.

G.f.: (x^3 - 2*x^2 - x + 1)/(1 - 2*x - 2*x^2 + 3*x^3 - x^5).

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

A322325 Number of nondecreasing Motzkin paths of length n.

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