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The Riordan square of the Motzkin numbers, triangle read by rows, T(n, k) for 0 <= k <= n.
1

%I #12 Mar 27 2020 13:44:30

%S 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,

%T 1,127,385,525,431,224,72,13,1,323,1030,1532,1429,889,369,98,15,1,835,

%U 2775,4428,4572,3269,1639,566,128,17,1

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

%e [0][ 1]

%e [1][ 1, 1]

%e [2][ 2, 3, 1]

%e [3][ 4, 8, 5, 1]

%e [4][ 9, 21, 18, 7, 1]

%e [5][ 21, 55, 58, 32, 9, 1]

%e [6][ 51, 145, 177, 123, 50, 11, 1]

%e [7][ 127, 385, 525, 431, 224, 72, 13, 1]

%e [8][ 323, 1030, 1532, 1429, 889, 369, 98, 15, 1]

%e [9][ 835, 2775, 4428, 4572, 3269, 1639, 566, 128, 17, 1]

%p # The function RiordanSquare is defined in A321620.

%p Motzkin := (1 - x - sqrt(1 - 2*x - 3*x^2))/(2*x^2); RiordanSquare(Motzkin, 10);

%t (* The function RiordanSquare is defined in A321620. *)

%t Motzkin = (1 - x - Sqrt[1 - 2 x - 3 x^2])/(2 x^2);

%t M = RiordanSquare[Motzkin, 10];

%t M // Flatten (* _Jean-François Alcover_, Nov 24 2018 *)

%o (Sage) # uses[riordan_square from A321620]

%o riordan_square((1 - x - sqrt(1 - 2*x - 3*x^2))/(2*x^2), 10)

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

%Y Cf. A321620.

%K tabl,nonn

%O 0,4

%A _Peter Luschny_, Nov 22 2018