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%I #28 Dec 14 2025 04:21:46
%S 1,1,1,1,4,1,1,9,10,1,1,16,42,20,1,1,25,120,140,35,1,1,36,275,600,378,
%T 56,1,1,49,546,1925,2310,882,84,1,1,64,980,5096,10010,7392,1848,120,1,
%U 1,81,1632,11760,34398,42042,20592,3564,165,1,1,100,2565,24480,99960,183456,150150,51480,6435,220,1
%N Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.
%C T(n,k) = number of operator monomials M in a ternary associative algebra with a unary linear operator L, where M has n total operations and exactly k occurrences of L. E.g., the T(n,k) = 10 such operator monomials are L(L(abc)), L(L(a)bc), L(L(a))bc, L(aL(b)c), L(abL(c)), L(a)L(b)c, L(a)bL(c), aL(L(b))c, aL(b)L(c), abL(L(c)).
%C T(n,k) = number of Schröder paths S of semilength 2n-k+1 with k up steps such that S has an odd number of horizontal (H) steps; and between each up step (that goes from y=i to y=i+1) and its matching down step (i.e., the next down step that goes from y=i+1 to y=i), there is an odd number of H steps.
%C E.g., the T(3,2) = 10 such paths are UUHHHDD, UUHDHHD, UUHDDHH, UHUHDHD, UHHDHDD, UHDUHDH, UHDHUHD, HUUHDDH, HUHDUHD, HHUUHDD.
%C T(n,k) = number of rooted ordered trees with 2n-k+1 edges and k+1 internal nodes where each internal node has odd outdegree.
%C T(n,k) = number of Dyck paths of semilength 2n-k+1 with k+1 peaks where every ascent (maximal run of up steps) has odd length. E.g., the T(3,2) = 10 such paths are UUUDUDUDDD, UUUDUDDUDD, UUUDUDDDUD, UUUDDUDUDD, UUUDDUDDUD, UUUDDDUDUD, UDUUUDUDDD, UDUUUDDUDD, UDUUUDDDUD, UDUDUUUDDD.
%C T(n,k) = number of 231-avoiding permutations of {1,2,...,2n-k+1} with k+1 decreasing runs, each run having odd length. E.g., the T(3,2) = 10 such permutations are 54123, 53124, 43125, 52134, 42135, 32145, 15423, 15324, 14325, 12543.
%C T(n,k) = number of Schröder paths of semilength n avoiding DDD and having exactly k total occurrences of H and DD. E.g., the T(3,2) = 10 such paths are UDHH, HUDH, HHUD, UUDDH, UHDH, HUUDD, HUHD, UHUDD, UUHDD, UHHD.
%C T(n,k) = number of labeled lattice paths from (0,0) with n steps of the form (l,1) with integer l>=0 such that the path stays weakly above y=x; every step (l,1) where l >=2 is labeled by a composition of l-1 consisting of 2 nonnegative parts; and exactly k of the n steps have l=0 (i.e., are vertical steps).
%C T(n,k) = number of labeled Dyck paths of semilength n+1 with k occurrences of UU, where every non-terminal descent of length l is assigned one of l labels (the final descent is unlabeled). E.g., the T(3,2) = 10 such paths are UUU(D)_1UDDD, UUU(DD)_1UDD, UUU(DD)_2UDD, UUU(DDD)_1UD, UUU(DDD)_2UD, UUU(DDD)_3UD, UU(D)_1UUDDD, UU(DD)_1UUDD, UU(DD)_2UUDD, U(D)_1UUUDDD.
%C T(n,k) = number of labeled rooted ordered trees with n+1 edges and k+1 leaves where each non-root internal node with outdegree l is assigned one of l labels.
%H Yu Hin Au and Murray R. Bremner, <a href="https://arxiv.org/abs/2511.13671">A new generalization of the Narayana numbers inspired by linear operators on associative d-ary algebras</a>, arXiv:2511.13671 [math.CO], 2025.
%H David Callan and Toufik Mansour, <a href="https://doi.org/10.1515/puma-2015-0027">Enumeration of small Wilf classes avoiding 1342 and two other 4-letter patterns</a>, Pure Mathematics and Applications (2018) Vol. 27, No. 1, 62-97. See Section 3.16.
%H JiSun Huh, Sangwook Kim, Seunghyun Seo, and Heesung Shin, <a href="https://doi.org/10.1016/j.aam.2024.102771">Bijections on pattern avoiding inversion sequences and related objects</a>, Advances in Applied Mathematics (2024) Vol. 161, Paper No. 102771.
%F G.f. u satisfies u = (1+x*y*u)*(1+x*u*(1+x*y*u)).
%e The initial rows of the triangle are:
%e [0] 1
%e [1] 1, 1
%e [2] 1, 4, 1
%e [3] 1, 9, 10, 1
%e [4] 1, 16, 42, 20, 1
%e [5] 1, 25, 120, 140, 35, 1
%e [6] 1, 36, 275, 600, 378, 56, 1
%e [7] 1, 49, 546, 1925, 2310, 882, 84, 1
%e [8] 1, 64, 980, 5096, 10010, 7392, 1848, 120, 1
%t T[n_,k_]:=Binomial[n+1,k+1]*Binomial[2n-k+1,k]/(n+1);Table[T[n,k],{n,0,10},{k,0,n}]//Flatten (* _James C. McMahon_, Dec 04 2025 *)
%Y Cf. A001263, A391046, A391047, A391048.
%Y Row sums give A106228(n+1).
%K easy,tabl,nonn
%O 0,5
%A _Yu Hin Au_, Nov 26 2025