A126322 Number of hex trees with n edges and no branches of length 1.

1, 0, 9, 27, 90, 297, 1053, 3888, 14742, 56619, 219429, 857304, 3375999, 13391001, 53452467, 214525017, 865041606, 3502806363, 14237599635, 58069495188, 237583710549, 974819569095, 4010205424869, 16536842688267, 68344258564980

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

G. C. Greubel, Table of n, a(n) for n = 0..1000

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

Formula

a(n) = A126321(n,0).

G.f.: (1-3z+9z^2)[1-3z-sqrt(1-6z+9z^2-36z^4)]/(18z^4).

Conjecture: (n+4)*(25*n^2+230*n+1137)*a(n) +3*(-50*n^3-585*n^2-3169*n-4248) *a(n-1) +9*(25*n^3+255*n^2+932*n-1764) *a(n-2) +29484*a(n-3) -36*(n-4) *(25*n^2+280*n+1392) *a(n-4)=0. - R. J. Mathar, Jun 17 2016

Maple

g:=(1-3*z+9*z^2)*(1-3*z-sqrt((1-3*z)^2-36*z^4))/18/z^4: gser:=series(g, z=0, 32): seq(coeff(gser, z, n), n=0..27);

Mathematica

CoefficientList[Series[(1 - 3*x + 9*x^2)*(1 - 3*x - Sqrt[1 - 6*x + 9*x^2 - 36*x^4])/(18*x^4), {x, 0, 30}], x] (* G. C. Greubel, Oct 23 2018 *)

Prog

(PARI) x='x+O('x^30); Vec((1-3*x+9*x^2)*(1-3*x-sqrt(1-6*x+9*x^2-36*x^4) )/(18*x^4)) \\ G. C. Greubel, Oct 23 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1 -3*x+9*x^2)*(1-3*x -Sqrt(1-6*x+9*x^2-36*x^4))/(18*x^4))); // G. C. Greubel, Oct 23 2018

Crossrefs

Cf. A126321.

Sequence in context: A255623 A036317 A053762 * A020279 A339725 A328604

Adjacent sequences: A126319 A126320 A126321 * A126323 A126324 A126325

Keyword

nonn

Author

Emeric Deutsch, Dec 25 2006

Status

approved