A120982 Triangle read by rows: T(n,k) is the number of ternary trees with n edges and having k vertices of outdegree 2 (n >= 0, k >= 0).

1, 3, 9, 3, 28, 27, 93, 162, 18, 333, 825, 270, 1272, 3915, 2430, 135, 5085, 18144, 17199, 2835, 20925, 84000, 106596, 34020, 1134, 87735, 391554, 612360, 308448, 30618, 372879, 1838295, 3369600, 2364390, 459270, 10206, 1602450, 8674380

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(2,1)=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.
Triangle starts:
    1;
    3;
    9,   3;
   28,  27;
   93, 162,  18;
  333, 825, 270;
		

Crossrefs

Cf. A001764, A003408, A120429, A120981, A120983, A120985.

Programs

Maple

T:=(n,k)->(1/(n+1))*binomial(n+1,k)*sum(3^(n-k-3*q)*binomial(n+1-k,k+1+2*q)*binomial(n-2*k-2*q,q),q=0..n/2-k):for n from 0 to 12 do seq(T(n,k),k=0..floor(n/2)) od; # yields sequence in triangular form

Formula

T(n,k) = (1/(n+1))*binomial(n+1,k)*Sum_{j=0..floor(n/2)-k} 3^(n-k-3j)*binomial(n+1-k, k+1+2j)*binomial(n-2k-2j, j).

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

Row n has 1+floor(n/2) terms.

Row sums yield A001764.

T(n,0) = A120985(n).

Sum_{k>=1} k*T(n,k) = 3*binomial(3n,n-2) = 3*A003408(n-2).