0,2

A hex tree is a rooted tree where each vertex has 0, 1, or 2 children and, when only one child is present, it is either a left child, or a middle child, or a right child (name due to an obvious bijection with certain tree-like polyhexes; see the Harary-Read reference).

Row n has floor(n/2) terms (n>=2).

Sum of terms in row n = A002212(n+1).

T(n,0) = A126189(n).

Sum_{k=0..floor(n/2)-1} k*T(n,k) = A126190(n).

F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.

G.f.: G = G(t,z) = 1+3*z*G+z^2*(1+3*z*G+t*(G-1-3*z*G))^2 (explicit expression in the Maple program).

Triangle starts:

1;

3;

10;

36;

135, 2;

519, 24;

2034, 180, 5;

8100, 1110, 75;

G:=1/2*(12*z^3*t+2*z^2*t^2-2*z^2*t-6*z^3*t^2-3*z-6*z^3+1-sqrt(1+9*z^2-4*z^2*t-6*z+12*z^3*t-12*z^3))/z^2/(3*z*t-t-3*z)^2: Gser:=simplify(series(G, z=0, 18)): for n from 0 to 14 do P[n]:=sort(coeff(Gser, z, n)) od: 1; 3; for n from 2 to 14 do seq(coeff(P[n], t, j), j=0..floor(n/2)-1) od;

g[t_, z_] = G /. Solve[G == 1 + 3z*G + z^2*(1 + 3z*G + t*(G - 1 - 3z*G))^2, G][[1]]; Flatten[ CoefficientList[ CoefficientList[ Series[g[t, z], {z, 0, 13}], z], t]][[1 ;; 39]] (* Jean-François Alcover, May 27 2011, after g.f. *)

Cf. A002212, A126189, A126190.

Sequence in context: A119374 A371773 A272686 * A081909 A371873 A126189

Adjacent sequences: A126185 A126186 A126187 * A126189 A126190 A126191

nonn,tabf

Emeric Deutsch, Dec 25 2006

approved