A126188 Triangle read by rows: T(n,k) is the number of hex trees with n edges and k pairs of adjacent vertices of outdegree 2.

1, 3, 10, 36, 135, 2, 519, 24, 2034, 180, 5, 8100, 1110, 75, 32688, 6210, 675, 14, 133380, 32886, 4851, 252, 549342, 168210, 30996, 2646, 42, 2280690, 840132, 184842, 21672, 882, 9534591, 4124682, 1053486, 154980, 10584, 132, 40103019

Offset: 0

Author

Emeric Deutsch, Dec 25 2006

Keywords

• nonn
• tabf

Comments

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).

Examples

			Triangle starts:
     1;
     3;
    10;
    36;
   135,    2;
   519,   24;
  2034,  180,    5;
  8100, 1110,   75;
		

Links

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

Crossrefs

Cf. A002212, A126189, A126190.

Programs

Maple

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;

Mathematica

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. *)

Formula

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).