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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).

%I A322378 #16 Feb 25 2020 11:26:19
%S A322378 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,
%T A322378 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,
%U A322378 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
%N 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).
%H A322378 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 A322378 Riordan array: ((1 - 2*x^2)/(1 - 3*x^2 + x^4), (x*(1-x^2))/(1 - 2*x^2)).
%e A322378 Triangle begins:
%e A322378    1;
%e A322378    0,   1;
%e A322378    1,   0,   1;
%e A322378    0,   2,   0,   1;
%e A322378    2,   0,   3,   0,   1;
%e A322378    0,   5,   0,   4,   0,   1;
%e A322378    5,   0,   9,   0,   5,   0,   1;
%e A322378    0,  13,   0,  14,   0,   6,   0,   1;
%e A322378   13,   0,  26,   0,  20,   0,   7,   0,   1;
%e A322378    0,  34,   0,  45,   0,  27,   0,   8,   0,   1;
%e A322378   34,   0,  73,   0,  71,   0,  35,   0,   9,   0,   1;
%e A322378    0,  89,   0, 137,   0, 105,   0,  44,   0,  10,   0,   1;
%e A322378   89,   0, 201,   0, 234,   0, 148,   0,  54,   0,  11,   0,   1;
%e A322378    0, 233,   0, 402,   0, 373,   0, 201,   0,  65,   0,  12,   0,   1;
%e A322378   ...
%Y A322378 Columns k=0, 1 give A001519. Column k=2 gives A061667.
%Y A322378 Cf. A322329, A322325.
%K A322378 nonn,tabl
%O A322378 0,8
%A A322378 _José Luis Ramírez Ramírez_, Dec 05 2018