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A322329 Triangle read by rows: T(n,k) is the number of nondecreasing Motzkin prefixes (i.e., left factors of nondecreasing Motzkin paths) of length n and final height k (0 <= k <= n).

%I A322329 #22 Dec 04 2018 16:57:53
%S A322329 1,1,1,2,2,1,4,5,3,1,9,12,9,4,1,21,30,25,14,5,1,49,74,69,44,20,6,1,
%T A322329 115,182,185,133,70,27,7,1,269,444,488,386,230,104,35,8,1,630,1078,
%U A322329 1266,1090,718,369,147,44,9,1,1474,2605,3245,3006,2161,1232,560,200,54,10,1
%N A322329 Triangle read by rows: T(n,k) is the number of nondecreasing Motzkin prefixes (i.e., left factors of nondecreasing Motzkin paths) of length n and final height k (0 <= k <= n).
%H A322329 R. Flórez and J. L. Ramírez, <a href="https://ajc.maths.uq.edu.au/pdf/72/ajc_v72_p138.pdf">Some enumerations on non-decreasing Motzkin paths</a>, Australasian Journal of Combinatorics, 72(1) (2018), 138-154.
%F A322329 Riordan array: ((1 - x - 2*x^2 + x^3)/(1 - 2*x - 2*x^2 + 3x^3 - x^5),(x*(1-x)^2*(1+x))/(1 - 2*x - x^2 + 2*x^3 - x^4)).
%e A322329 Triangle begins:
%e A322329      1;
%e A322329      1,    1;
%e A322329      2,    2,    1;
%e A322329      4,    5,    3,    1;
%e A322329      9,   12,    9,    4,    1;
%e A322329     21,   30,   25,   14,    5,    1;
%e A322329     49,   74,   69,   44,   20,    6,   1;
%e A322329    115,  182,  185,  133,   70,   27,   7,   1;
%e A322329    269,  444,  488,  386,  230,  104,  35,   8,  1;
%e A322329    630, 1078, 1266, 1090,  718,  369, 147,  44,  9,  1;
%e A322329   1474, 2605, 3245, 3006, 2161, 1232, 560, 200, 54, 10, 1;
%e A322329   ...
%Y A322329 Column k=0 gives A322325.
%Y A322329 Cf. A097862, A283595.
%K A322329 nonn,tabl
%O A322329 0,4
%A A322329 _José Luis Ramírez Ramírez_, Dec 03 2018