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Triangle read by rows: T(n,k) is the number of ternary trees with n edges and path length k; 0<=k<=n(n-1)/2.
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%I #2 Mar 30 2012 18:37:20

%S 1,1,0,3,0,0,3,9,0,0,0,1,18,9,27,0,0,0,0,0,9,45,57,54,27,81,0,0,0,0,0,

%T 0,0,36,87,270,81,297,171,162,81,243,0,0,0,0,0,0,0,0,0,84,261,567,756,

%U 936,585,972,729,891,513,486,243,729,0,0,0,0,0,0,0,0,0,0,0,126,774,1080

%N Triangle read by rows: T(n,k) is the number of ternary trees with n edges and path length k; 0<=k<=n(n-1)/2.

%F G.f. satisfies: A(x,q) = 1 + x*A(q*x,q)^3.

%F Row sums equal A001764, which enumerates ternary trees and has g.f.: G(x) = 1 + x*G(x)^3.

%F Column sums equal A132331(k), which is the number of ternary trees of path length k.

%e G.f.: A(x,q) = 1 + x + (3*q)*x^2 + (3*q^2 + 9*q^3)*x^3 + (q^3 + 18*q^4 + 9*q^5 + 27*q^6)*x^4 +...

%e A(x,q)^3 = 1 + 3*x + (3 + 9*q)*x^2 + (1 + 18*q + 9*q^2 + 27*q^3)*x^3 +...

%e Triangle begins:

%e 1;

%e 1;

%e 0,3;

%e 0,0,3,9;

%e 0,0,0,1,18,9,27;

%e 0,0,0,0,0,9,45,57,54,27,81;

%e 0,0,0,0,0,0,0,36,87,270,81,297,171,162,81,243;

%e 0,0,0,0,0,0,0,0,0,84,261,567,756,936,585,972,729,891,513,486,243,729;

%e 0,0,0,0,0,0,0,0,0,0,0,126,774,1080,2817,2682,4383,1998,4941,3294,3780,2241,4374,2187,2673,1539,1458,729,2187; ...

%o (PARI) {T(n,k)=local(A=1+x);for(i=1,n,A=1+x*subst(A,x,q*x+x*O(x^n))^3); polcoeff(polcoeff(A,n,x)+O(q^(n*(n-1)/2+1)),k,q)}

%Y Cf. A001764 (row sums), A132331 (column sums), A138157 (variant).

%K nonn,tabl

%O 0,4

%A _Paul D. Hanna_, Jan 29 2010