A120983 Triangle read by rows: T(n,k) is the number of ternary trees with n edges and having k vertices of outdegree 3 (n >= 0, k >= 0).

1, 3, 12, 54, 1, 261, 12, 1323, 105, 6939, 810, 3, 37341, 5859, 63, 205011, 40824, 840, 1143801, 277830, 9072, 12, 6466230, 1861380, 86670, 360, 36960300, 12335895, 764478, 6435, 213243435, 81120204, 6377778, 89100, 55, 1240219269, 530408736

Offset: 0

Author

Emeric Deutsch, Jul 21 2006

Keywords

• nonn
• tabf

Comments

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.

Examples

			T(3,1)=1 because we have (Q,L,M,R), where Q denotes the root and L (M,R) denotes a left (middle, right) child of Q.
Triangle starts:
     1;
     3;
    12;
    54,   1;
   261,  12;
  1323, 105;
  6939, 810, 3;
		

Crossrefs

Cf. A001764, A107264, A120429, A120981, A120982, A004321.

Programs

Maple

T:=(n,k)->(1/(n+1))*binomial(n+1,k)*sum(3^j*binomial(n+1-k,j)*binomial(j,n-3*k-j),j=0..n+1-k): for n from 0 to 14 do seq(T(n,k),k=0..floor(n/3)) od; # yields sequence in triangular form

Formula

T(n,k) = (1/(n+1))*binomial(n+1,k)*Sum_{j=0..n+1-k} 3^j*binomial(n+1-k, j)*binomial(j, n-3k-j).

G.f.: G = G(t,z) satisfies G = 1 + 3zG + 3z^2*G^2 + tz^3*G^3.

Row n has 1+floor(n/3) terms.

Row sums yield A001764.

T(n,0) = A107264(n).

Sum_{k>=1} k*T(n,k) = binomial(3n, n-3) = A004321(n).