A120983 - OEIS

%I #9 Nov 16 2019 10:32:35

%S 1,3,12,54,1,261,12,1323,105,6939,810,3,37341,5859,63,205011,40824,

%T 840,1143801,277830,9072,12,6466230,1861380,86670,360,36960300,

%U 12335895,764478,6435,213243435,81120204,6377778,89100,55,1240219269,530408736

%N 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).

%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.

%F 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).

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

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

%F Row sums yield A001764.

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

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

%e 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.

%e Triangle starts:

%e      1;

%e      3;

%e     12;

%e     54,   1;

%e    261,  12;

%e   1323, 105;

%e   6939, 810, 3;

%p 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

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

%K nonn,tabf

%O 0,2

%A _Emeric Deutsch_, Jul 21 2006