A322329 -id:A322329

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A322325

Number of nondecreasing Motzkin paths of length n.

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..32.

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)

FORMULA

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).

EXAMPLE

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.

MATHEMATICA

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

CROSSREFS

Column k=0 of A322329.

Cf. A001006, A001519.

KEYWORD

nonn,easy

AUTHOR

José Luis Ramírez Ramírez, Dec 03 2018

STATUS

approved

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).

+10
0

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

(list; table; graph; refs; listen; history; text; internal format)

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:

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.

KEYWORD

nonn,tabl

AUTHOR

José Luis Ramírez Ramírez, Dec 05 2018

STATUS

approved

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