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

0,2

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).

Links

Table of n, a(n) for n=0..38.

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

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).

Example

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

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. *)

Crossrefs

Cf. A002212, A126189, A126190.

Sequence in context: A119374 A371773 A272686 * A081909 A371873 A126189

Adjacent sequences: A126185 A126186 A126187 * A126189 A126190 A126191

Keyword

nonn,tabf

Author

Emeric Deutsch, Dec 25 2006

Status

approved