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%I #22 Dec 29 2016 02:29:17
%S 1,3,10,33,3,108,29,351,186,6,1134,990,95,3645,4725,900,15,11664,
%T 20979,6615,329,37179,88452,41580,4116,42,118098,358668,234738,38556,
%U 1176,373977,1410615,1224720,300510,18270,126,1180980,5412825,6014250
%N Triangle read by rows: T(n,k) is number of hex trees with n edges and k nonroot nodes of outdegree 2.
%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).
%C Row 0 has one term; rows 2n-1 and 2n have n terms.
%C Sum of terms in row n = A002212(n+1).
%C T(n,0)=A126184(n).
%C Sum_{k=1..floor((n-1)/2)} k*T(n,k) = A126185(n).
%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 G.f.: G(t,z)=1+3*z*H+z^2*H^2, where H=H(t,z) is defined by H=1+3*z*H+t*z^2*H^2 (see explicit expression of G(t,z) at the Maple program).
%e Triangle begins:
%e 1;
%e 3;
%e 10;
%e 33, 3;
%e 108, 29;
%e 351, 186, 6;
%p G := 1/2/t^2/z^2*(-11*t*z^2+2*t^2*z^2+3*z*t+9*z^2-6*z+1-sqrt(1-58*t*z^2-12*z+54*z^2 +6*z*t+81*z^4-108*z^3 -36*t^3*z^4+153*t^2*z^4 -198*t*z^4-78*t^2*z^3+186*t*z^3+9*t^2*z^2)): Gser:=simplify(series(G,z=0,16)): for n from 0 to 18 do P[n]:=sort(coeff(Gser,z,n)) od: 1; for n from 1 to 13 do seq(coeff(P[n],t,j),j=0..floor((n-1)/2)) od; # yields sequence in triangular form
%t len = 40; m = Ceiling[2 Sqrt[len]]; gf[t_, z_] = g /. Solve[g == 1 + 3z* h + z^2*h^2 && h == 1 + 3z*h + t*z^2*h^2, g, h][[1]]; gser = Series[gf[t, z], {z, 0, m}]; p[n_] := Coefficient[gser, z, n]; tr[n_, k_] := tr[n, k] = Coefficient[p[n], t, k]; Flatten[Table[ tr[n, k], {n, 0, m}, {k, 0, Max[0, Floor[(n-1)/2]]}]][[1 ;; len]] (* _Jean-François Alcover_, May 31 2011, after Maple prog. *)
%Y Cf. A002212, A126184, A126185.
%K nonn,tabf
%O 0,2
%A _Emeric Deutsch_, Dec 19 2006
%E Keyword tabl changed to tabf by _Michel Marcus_, Apr 09 2013
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