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%I #64 Jun 04 2022 01:56:17
%S 1,2,1,5,6,1,14,28,12,1,42,120,90,20,1,132,495,550,220,30,1,429,2002,
%T 3003,1820,455,42,1,1430,8008,15288,12740,4900,840,56,1,4862,31824,
%U 74256,79968,42840,11424,1428,72,1,16796,125970,348840,465120,325584
%N Triangle read by rows: T(n,k) is the number of ternary trees with n edges and having k middle edges (n >= 0, k >= 0).
%C A ternary tree is a rooted tree in which each vertex has at most three children and each child of a vertex is designated as its left or middle or right child.
%C T(n,k) is the number of Dyck paths of semilength 2n+2 with all descent runs of even length and n+1-k peaks. - _Alexander Burstein_, May 23 2020
%C T(n,k) is the number of Dyck paths of semilength 2n+2 with all descent runs of even length and k+1 peaks at even height. - _Alexander Burstein_, Jun 03 2020
%C T(n,k) is the number of Dyck paths of semilength 2n+2 with all descent runs of even length and k peaks at odd height. - _Alexander Burstein_, Jun 18 2020
%C An apparent refinement is A338135. - _Tom Copeland_, Oct 12 2020
%H Andrew Howroyd, <a href="/A120986/b120986.txt">Table of n, a(n) for n = 0..1274</a>
%H Alexander Burstein, <a href="https://arxiv.org/abs/2009.00760">Distribution of peak heights modulo k and double descents on k-Dyck paths</a>, arXiv:2009.00760 [math.CO], 2020.
%H Alexander Burstein and Megan Martinez, <a href="https://permutationpatterns.com/files/2020/06/WednesdayA_Burstein.pdf">Pattern classes equinumerous to the class of ternary forests</a>, Permutation Patterns Virtual Workshop, Howard University (2020).
%H Helmut Prodinger, <a href="https://arxiv.org/abs/2009.06793">Counting ternary trees according to the number of middle edges and factorizing into (3/2)-ary trees</a>, arXiv:2009.06793 [math.CO], 2020.
%H Helmut Prodinger, <a href="https://arxiv.org/abs/2205.13374">Counting edges according to edge-type in t-ary trees</a>, arXiv:2205.13374 [math.CO], 2022.
%H Chao-Jen Wang, <a href="http://people.brandeis.edu/~gessel/homepage/students/wangthesis.pdf">Applications of the Goulden-Jackson cluster method to counting Dyck paths by occurrences of subwords</a>, Dissertation, Brandeis University, 2011.
%F T(n,k) = (1/(n+1))*binomial(n+1,k)*binomial(2*(n+1),n-k).
%F T(n,0) = A000108(n+1) (the Catalan numbers).
%F T(n,k) = A108767(n+1,n+1-k).
%F Sum_{k>=1} k*T(n,k) = binomial(3*n+2,n-1) = A013698(n).
%F G.f.: G = G(t,z) satisfies G = (1+t*z*G)(1+z*G)^2.
%F O.g.f.: A(x,t) = 1 + (2 + t)*x + (5 + 6*t + t^2)*x^2 + ... satisfies 1 + x*d/dx(A(x,t))/A(x,t) = 1 + (2 + t)*x + (6 + 8*t + t^2)*x^2 + ..., which is the o.g.f. for A110608. - _Peter Bala_, Oct 13 2015
%e Triangle starts:
%e 1;
%e 2, 1;
%e 5, 6, 1;
%e 14, 28, 12, 1;
%e 42, 120, 90, 20, 1;
%e 132, 495, 550, 220, 30, 1;
%e ...
%p T:=(n,k)->binomial(n+1,k)*binomial(2*(n+1),n-k)/(n+1): for n from 0 to 10 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
%t T[n_, k_] := Binomial[n+1, k]*Binomial[2*(n+1), n-k]/(n+1);
%t Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 06 2018 *)
%o (PARI) T(n,k) = binomial(n+1,k)*binomial(2*(n+1),n-k)/(n+1); \\ _Andrew Howroyd_, Nov 06 2017
%o (Python)
%o from sympy import binomial
%o def T(n, k): return binomial(n + 1, k)*binomial(2*(n + 1), n - k)//(n + 1)
%o for n in range(21): print([T(n, k) for k in range(n + 1)]) # _Indranil Ghosh_, Nov 07 2017
%Y Cf. A001764 (row sums), A000108, A108767, A013698, A110608.
%K nonn,tabl
%O 0,2
%A _Emeric Deutsch_, Jul 21 2006
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