A126180 Number of branches in all hex trees with n edges (n>=1).

3, 11, 48, 224, 1071, 5169, 25053, 121711, 592233, 2885397, 14073318, 68710266, 335775825, 1642305765, 8039194560, 39382567940, 193067419905, 947129136345, 4649300253960, 22836432229240, 112231899902085, 551871446928895

Offset

1,1

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 median 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=1..22.

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

J. Riordan, Enumeration of plane trees by branches and endpoints, J. Comb. Theory (A) 19, 1975, 214-222.

Formula

a(n) = Sum_{k=1..n} k*A126179(n,k).

G.f.: (1-3z)[2-9z+5z^2-(2-3z)sqrt(1-6z+5z^2)]/[2z^2*sqrt(1-6z+5z^2)].

Conjecture: -2*(n+2)*(239*n-1252)*a(n) +21*(220*n^2-997*n-578)*a(n-1) +(-12902*n^2+76676*n-77157)*a(n-2) +15*(n-3)*(584*n-2237)*a(n-3)=0. - R. J. Mathar, Jun 17 2016

Example

a(2)=11 because there are 3^2=9 path-trees of length 2 (each has 1 branch) and one V-shaped tree having 2 branches.

Maple

G:=(1-3*z)*(2-9*z+5*z^2-(2-3*z)*sqrt(1-6*z+5*z^2))/2/z^2/sqrt(1-6*z+5*z^2): Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=1..26);

Crossrefs

Cf. A126179.

Sequence in context: A191344 A308538 A181278 * A121139 A316703 A362741

Adjacent sequences: A126177 A126178 A126179 * A126181 A126182 A126183

Keyword

nonn

Author

Emeric Deutsch, Dec 19 2006

Status

approved