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A126322 Number of hex trees with n edges and no branches of length 1. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 2 08:31 EST 2021. Contains 349437 sequences. (Running on oeis4.)