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A120981 Triangle read by rows: T(n,k) is the number of ternary trees with n edges and having k vertices of outdegree 1 (n >= 0, k >= 0).

%I A120981 #20 Jul 02 2021 16:46:45
%S A120981 1,0,3,3,0,9,1,27,0,27,18,12,162,0,81,15,270,90,810,0,243,138,270,
%T A120981 2430,540,3645,0,729,189,2898,2835,17010,2835,15309,0,2187,1218,4536,
%U A120981 34776,22680,102060,13608,61236,0,6561,2280,32886,61236,312984,153090,551124
%N A120981 Triangle read by rows: T(n,k) is the number of ternary trees with n edges and having k vertices of outdegree 1 (n >= 0, k >= 0).
%C A120981 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.
%H A120981 Andrew Howroyd, <a href="/A120981/b120981.txt">Table of n, a(n) for n = 0..1274</a>
%H A120981 Paul Barry, <a href="https://arxiv.org/abs/2104.01644">Centered polygon numbers, heptagons and nonagons, and the Robbins numbers</a>, arXiv:2104.01644 [math.CO], 2021.
%F A120981 T(n,0) = A120984(n).
%F A120981 Sum_{k>=1} k*T(n,k) = 3*binomial(3n,n-1) = 3*A004319(n).
%F A120981 T(n,k) = (1/(n+1))*binomial(n+1,k)*Sum_{j=0..n+1-k} 3^(2k-n+3j)*binomial(n+1-k,j)*binomial(j,n-k-2j).
%F A120981 G.f.: G=G(t,z) satisfies G = 1 + 3tzG + 3z^2*G^2 + z^3*G^3.
%e A120981 T(2,0)=3 because we have (Q,L,M), (Q,L,R) and (Q,M,R), where Q denotes the root and L (M,R) denotes a left (middle, right) child of Q.
%e A120981 Triangle starts:
%e A120981    1;
%e A120981    0,   3;
%e A120981    3,   0,   9;
%e A120981    1,  27,   0,  27;
%e A120981   18,  12, 162,   0, 81;
%e A120981   15, 270,  90, 810,  0, 243;
%p A120981 T:=proc(n,k) if k<=n then (1/(n+1))*binomial(n+1,k)*sum(3^(3*j-n+2*k)*binomial(n+1-k,j)*binomial(j,n-k-2*j),j=0..n+1-k) else 0 fi end: for n from 0 to 10 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
%t A120981 T[n_, k_] := (1/(n+1))*Binomial[n+1, k]*Sum[3^(2k - n + 3j)*Binomial[n + 1 - k, j]*Binomial[j, n - k - 2j], {j, 0, n - k + 1}];
%t A120981 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 02 2018 *)
%o A120981 (PARI) T(n,k) = binomial(n+1, k)*sum(j=0, n+1-k, 3^(2*k-n+3*j)*binomial(n+1-k, j)*binomial(j, n-k-2*j))/(n+1); \\ _Andrew Howroyd_, Nov 06 2017
%o A120981 (Python)
%o A120981 from sympy import binomial
%o A120981 def T(n, k): return binomial(n + 1, k)*sum([3**(2*k - n + 3*j)*binomial(n + 1 - k, j)*binomial(j, n - k - 2*j) for j in range(n + 2 - k)])//(n + 1)
%o A120981 for n in range(21): print([T(n, k) for k in range(n + 1)]) # _Indranil Ghosh_, Nov 07 2017
%Y A120981 Diagonals include A129530, A036216.
%Y A120981 Cf. A001764 (row sums), A004319, A120429, A120982, A120983, A120984.
%K A120981 nonn,tabl
%O A120981 0,3
%A A120981 _Emeric Deutsch_, Jul 21 2006