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%I #14 Dec 30 2016 02:30:41
%S 3,11,48,224,1071,5169,25053,121711,592233,2885397,14073318,68710266,
%T 335775825,1642305765,8039194560,39382567940,193067419905,
%U 947129136345,4649300253960,22836432229240,112231899902085,551871446928895
%N Number of branches in all hex trees with n edges (n>=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 median child, or a right child (name due to an obvious bijection with certain tree-like polyhexes; see the Harary-Read reference).
%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.
%H J. Riordan, <a href="http://dx.doi.org/10.1016/S0097-3165(75)80010-0">Enumeration of plane trees by branches and endpoints</a>, J. Comb. Theory (A) 19, 1975, 214-222.
%F a(n) = Sum_{k=1..n} k*A126179(n,k).
%F G.f.: (1-3z)[2-9z+5z^2-(2-3z)sqrt(1-6z+5z^2)]/[2z^2*sqrt(1-6z+5z^2)].
%F 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
%e 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.
%p 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);
%Y Cf. A126179.
%K nonn
%O 1,1
%A _Emeric Deutsch_, Dec 19 2006
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