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 A126183 Triangle read by rows: T(n,k) is number of hex trees with n edges and k nonroot nodes of outdegree 2. 2
 1, 3, 10, 33, 3, 108, 29, 351, 186, 6, 1134, 990, 95, 3645, 4725, 900, 15, 11664, 20979, 6615, 329, 37179, 88452, 41580, 4116, 42, 118098, 358668, 234738, 38556, 1176, 373977, 1410615, 1224720, 300510, 18270, 126, 1180980, 5412825, 6014250 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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). Row 0 has one term; rows 2n-1 and 2n have n terms. Sum of terms in row n = A002212(n+1). T(n,0)=A126184(n). Sum_{k=1..floor((n-1)/2)} k*T(n,k) = A126185(n). LINKS F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13. FORMULA 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). EXAMPLE Triangle begins:     1;     3;    10;    33,   3;   108,  29;   351, 186,   6; MAPLE 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 MATHEMATICA 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. *) CROSSREFS Cf. A002212, A126184, A126185. Sequence in context: A184436 A149028 A174573 * A174798 A042039 A080697 Adjacent sequences:  A126180 A126181 A126182 * A126184 A126185 A126186 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Dec 19 2006 EXTENSIONS Keyword tabl changed to tabf by Michel Marcus, Apr 09 2013 STATUS approved

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Last modified May 7 22:52 EDT 2021. Contains 343652 sequences. (Running on oeis4.)