A126321 Triangle read by rows: number of hex trees with n edges and k branches of length 1 (0<=k<=n).

1, 0, 3, 9, 0, 1, 27, 6, 0, 3, 90, 18, 27, 0, 2, 297, 135, 81, 24, 0, 6, 1053, 648, 351, 72, 90, 0, 5, 3888, 2889, 1377, 756, 270, 90, 0, 15, 14742, 12150, 6723, 3888, 1485, 270, 315, 0, 14, 56619, 51273, 32805, 19224, 6480, 3645, 945, 336, 0, 42, 219429, 218700

Offset

0,3

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).

Links

Table of n, a(n) for n=0..56.

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

Formula

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

T(n,0) = A126322(n). T(n,n) = A126324(n).

T(2n,2n) = Cat(n); T(2n+1,2n+1) = 3*Cat(n), where Cat(n) are the Catalan numbers (A000108).

Sum_{k=0..n} k*T(n,k) = A126323(n).

G.f.: (1+P)C(z^2*Q^2), where C(z)=(1-sqrt(1-4z))/(2z) is the Catalan function, P=3tz + 9z^2/(1-3z) and Q=t+3z/(1-3z).

Example

T(2,2)=1 because among the 10 hex trees with two edges only the tree V has 2 branches of length 1.
Triangle starts:
    1;
    0,   3;
    9,   0,   1;
   27,   6,   0,   3;
   90,  18,  27,   0,   2;
  297, 135,  81,  24,   0,   6;

Maple

C:=z->(1-sqrt(1-4*z))/2/z: G:=(1+3*t*z+9*z^2/(1-3*z))*C(z^2*(t+3*z/(1-3*z))^2): Gser:=simplify(series(G, z=0, 15)): for n from 0 to 10 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 10 do seq(coeff(P[n], t, j), j=0..n) od; # yields sequence in triangular form

Mathematica

n = 10; c[z_] = (1 - Sqrt[1 - 4*z])/2/z; g[t_, z_] = (1 + 3*t*z + 9*(z^2/(1 - 3*z)))* c[z^2*(t + 3*(z/(1 - 3*z)))^2]; Flatten[CoefficientList[#, t] & /@ CoefficientList[Series[g[t, z], {z, 0, n}], z]] (* Jean-François Alcover, Jul 22 2011, after Maple prog. *)

Crossrefs

Cf. A000108, A002212, A126322, A126323, A126324.

Sequence in context: A329937 A272535 A016626 * A335777 A358945 A248726

Adjacent sequences: A126318 A126319 A126320 * A126322 A126323 A126324

Keyword

nonn,tabl

Author

Emeric Deutsch, Dec 25 2006

Status

approved