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%I #13 Sep 08 2022 08:45:29
%S 1,0,9,27,90,297,1053,3888,14742,56619,219429,857304,3375999,13391001,
%T 53452467,214525017,865041606,3502806363,14237599635,58069495188,
%U 237583710549,974819569095,4010205424869,16536842688267,68344258564980
%N Number of hex trees with n edges and no branches of length 1.
%C 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).
%H G. C. Greubel, <a href="/A126322/b126322.txt">Table of n, a(n) for n = 0..1000</a>
%H F. Harary and R. C. Read, <a href="https://doi.org/10.1017/S0013091500009135">The enumeration of tree-like polyhexes</a>, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.
%F a(n) = A126321(n,0).
%F G.f.: (1-3z+9z^2)[1-3z-sqrt(1-6z+9z^2-36z^4)]/(18z^4).
%F 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
%p 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);
%t 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 *)
%o (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
%o (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
%Y Cf. A126321.
%K nonn
%O 0,3
%A _Emeric Deutsch_, Dec 25 2006
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