A321621 The Riordan square of the Motzkin numbers, triangle read by rows, T(n, k) for 0 <= k <= n.

1, 1, 1, 2, 3, 1, 4, 8, 5, 1, 9, 21, 18, 7, 1, 21, 55, 58, 32, 9, 1, 51, 145, 177, 123, 50, 11, 1, 127, 385, 525, 431, 224, 72, 13, 1, 323, 1030, 1532, 1429, 889, 369, 98, 15, 1, 835, 2775, 4428, 4572, 3269, 1639, 566, 128, 17, 1

Offset: 0

Author

Peter Luschny, Nov 22 2018

Keywords

• tabl
• nonn

Examples

			[0][   1]
[1][   1,    1]
[2][   2,    3,    1]
[3][   4,    8,    5,    1]
[4][   9,   21,   18,    7,    1]
[5][  21,   55,   58,   32,    9,    1]
[6][  51,  145,  177,  123,   50,   11,   1]
[7][ 127,  385,  525,  431,  224,   72,  13,   1]
[8][ 323, 1030, 1532, 1429,  889,  369,  98,  15,  1]
[9][ 835, 2775, 4428, 4572, 3269, 1639, 566, 128, 17, 1]
		

Links

• Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 16, 29.

Crossrefs

T(n, 0) = A001006 (Motzkin), A111961 (row sums), A000007 (alternating row sums).

Cf. A321620.

Programs

Maple

# The function RiordanSquare is defined in A321620.
Motzkin := (1 - x - sqrt(1 - 2*x - 3*x^2))/(2*x^2); RiordanSquare(Motzkin, 10);

Mathematica

(* The function RiordanSquare is defined in A321620. *)
Motzkin = (1 - x - Sqrt[1 - 2 x - 3 x^2])/(2 x^2);
M = RiordanSquare[Motzkin, 10];
M // Flatten (* Jean-François Alcover, Nov 24 2018 *)

Sage

# uses[riordan_square from A321620]
riordan_square((1 - x - sqrt(1 - 2*x - 3*x^2))/(2*x^2), 10)