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A126188 Triangle read by rows: T(n,k) is the number of hex trees with n edges and k pairs of adjacent vertices of outdegree 2.

%I A126188 #17 Feb 06 2021 15:22:07
%S A126188 1,3,10,36,135,2,519,24,2034,180,5,8100,1110,75,32688,6210,675,14,
%T A126188 133380,32886,4851,252,549342,168210,30996,2646,42,2280690,840132,
%U A126188 184842,21672,882,9534591,4124682,1053486,154980,10584,132,40103019
%N A126188 Triangle read by rows: T(n,k) is the number of hex trees with n edges and k pairs of adjacent vertices of outdegree 2.
%C A126188 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).
%C A126188 Row n has floor(n/2) terms (n>=2).
%C A126188 Sum of terms in row n = A002212(n+1).
%C A126188 T(n,0) = A126189(n).
%C A126188 Sum_{k=0..floor(n/2)-1} k*T(n,k) = A126190(n).
%H A126188 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 A126188 G.f.: G = G(t,z) = 1+3*z*G+z^2*(1+3*z*G+t*(G-1-3*z*G))^2 (explicit expression in the Maple program).
%e A126188 Triangle starts:
%e A126188      1;
%e A126188      3;
%e A126188     10;
%e A126188     36;
%e A126188    135,    2;
%e A126188    519,   24;
%e A126188   2034,  180,    5;
%e A126188   8100, 1110,   75;
%p A126188 G:=1/2*(12*z^3*t+2*z^2*t^2-2*z^2*t-6*z^3*t^2-3*z-6*z^3+1-sqrt(1+9*z^2-4*z^2*t-6*z+12*z^3*t-12*z^3))/z^2/(3*z*t-t-3*z)^2: Gser:=simplify(series(G,z=0,18)): for n from 0 to 14 do P[n]:=sort(coeff(Gser,z,n)) od: 1;3;for n from 2 to 14 do seq(coeff(P[n],t,j),j=0..floor(n/2)-1) od;
%t A126188 g[t_,z_] = G /. Solve[G == 1 + 3z*G + z^2*(1 + 3z*G + t*(G - 1 - 3z*G))^2, G][[1]]; Flatten[ CoefficientList[ CoefficientList[ Series[g[t,z], {z,0,13}], z], t]][[1 ;; 39]] (* _Jean-François Alcover_, May 27 2011, after g.f. *)
%Y A126188 Cf. A002212, A126189, A126190.
%K A126188 nonn,tabf
%O A126188 0,2
%A A126188 _Emeric Deutsch_, Dec 25 2006