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A126186 Triangle read by rows: T(n,k) is number of hex trees with n edges and level of first leaf (in the preorder traversal) equal to k (1 <= k <= n).

%I A126186 #19 Jun 25 2019 01:56:59
%S A126186 3,1,9,3,6,27,10,19,27,81,36,66,90,108,243,137,245,325,378,405,729,
%T A126186 543,954,1242,1416,1485,1458,2187,2219,3848,4944,5563,5760,5589,5103,
%U A126186 6561,9285,15942,20286,22620,23235,22410,20412,17496,19683,39587,67442,85194
%N A126186 Triangle read by rows: T(n,k) is number of hex trees with n edges and level of first leaf (in the preorder traversal) equal to k (1 <= k <= n).
%C A126186 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 A126186 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 A126186 T(n,k) = [k/(n-k)] sum(3^(2k+2j-n)*binomial(n-k,j)*binomial(k-1+j,n-k-1-j), j=ceiling((n-2k)/2)..n-k) if 1<=k<n; T(n,n)=3^n.
%F A126186 G.f.: 2/[2-t-3tz+t*sqrt(1-6z+5z^2)]-1.
%F A126186 Sum of terms in row n = A002212(n+1).
%F A126186 T(n,1) = A025238(n); T(n,1) = A002212(n-1) for n>=2.
%F A126186 T(n,n) = 3^n = A000244(n); T(n,n-1) = (n-1)*3^(n-2) = A027471(n) (n>=2).
%F A126186 Sum_{k=1..n} k*T(n,k) = A126187(n).
%e A126186 Triangle starts:
%e A126186    3;
%e A126186    1,   9;
%e A126186    3,   6,  27;
%e A126186   10,  19,  27,  81;
%e A126186   36,  66,  90, 108, 243;
%p A126186 G:=2/(2-t-3*t*z+t*sqrt(1-6*z+5*z^2))-1: Gser:=simplify(series(G,z=0,14)): for n from 1 to 10 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 1 to 10 do seq(coeff(P[n],t,k),k=1..n) od; # yields sequence in triangular form
%t A126186 n = 10; g[t_, z_] = 2/(2 - t - 3t*z + t*Sqrt[1 - 6z + 5z^2]) - 1; Flatten[ Rest[ CoefficientList[#, t]] & /@ Rest[ CoefficientList[ Series[g[t, z], {z, 0, n}], z]]] (* _Jean-François Alcover_, Jul 22 2011, after g.f. *)
%Y A126186 Cf. A000244, A002212, A025238, A027471, A126187.
%K A126186 nonn,tabl
%O A126186 1,1
%A A126186 _Emeric Deutsch_, Dec 22 2006