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

%I A321621 #16 Apr 18 2025 13:12:35
%S A321621 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 A321621 1,127,385,525,431,224,72,13,1,323,1030,1532,1429,889,369,98,15,1,835,
%U A321621 2775,4428,4572,3269,1639,566,128,17,1
%N A321621 The Riordan square of the Motzkin numbers, triangle read by rows, T(n, k) for 0 <= k <= n.
%H A321621 Paul Barry, <a href="https://arxiv.org/abs/2504.09719">Notes on Riordan arrays and lattice paths</a>, arXiv:2504.09719 [math.CO], 2025. See pp. 16, 29.
%e A321621 [0][   1]
%e A321621 [1][   1,    1]
%e A321621 [2][   2,    3,    1]
%e A321621 [3][   4,    8,    5,    1]
%e A321621 [4][   9,   21,   18,    7,    1]
%e A321621 [5][  21,   55,   58,   32,    9,    1]
%e A321621 [6][  51,  145,  177,  123,   50,   11,   1]
%e A321621 [7][ 127,  385,  525,  431,  224,   72,  13,   1]
%e A321621 [8][ 323, 1030, 1532, 1429,  889,  369,  98,  15,  1]
%e A321621 [9][ 835, 2775, 4428, 4572, 3269, 1639, 566, 128, 17, 1]
%p A321621 # The function RiordanSquare is defined in A321620.
%p A321621 Motzkin := (1 - x - sqrt(1 - 2*x - 3*x^2))/(2*x^2); RiordanSquare(Motzkin, 10);
%t A321621 (* The function RiordanSquare is defined in A321620. *)
%t A321621 Motzkin = (1 - x - Sqrt[1 - 2 x - 3 x^2])/(2 x^2);
%t A321621 M = RiordanSquare[Motzkin, 10];
%t A321621 M // Flatten (* _Jean-François Alcover_, Nov 24 2018 *)
%o A321621 (Sage) # uses[riordan_square from A321620]
%o A321621 riordan_square((1 - x - sqrt(1 - 2*x - 3*x^2))/(2*x^2), 10)
%Y A321621 T(n, 0) = A001006 (Motzkin), A111961 (row sums), A000007 (alternating row sums).
%Y A321621 Cf. A321620.
%K A321621 tabl,nonn
%O A321621 0,4
%A A321621 _Peter Luschny_, Nov 22 2018