A036370 Triangle of coefficients of generating function of ternary rooted trees of height at most n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 5, 4, 4, 3, 2, 1, 1, 1, 1, 1, 2, 4, 7, 12, 20, 31, 47, 70, 99, 137, 184, 239, 300, 369, 432, 498, 551, 594, 614, 624, 601, 570, 514, 453, 378, 312, 238, 181, 128, 89, 56, 37, 20, 12, 6, 3, 1, 1

Offset: 0

Author

N. J. A. Sloane, Eric Rains (rains(AT)caltech.edu)

Keywords

• nonn
• easy
• tabf

Examples

			1;
1, 1;
1, 1, 1, 1, 1;
1, 1, 1, 2, 3, 4, 4, 5, 4, 4, 3, 2, 1, 1;
...
		

Links

• Alois P. Heinz, Rows n = 0..8, flattened
• Index entries for sequences related to rooted trees

Crossrefs

Cf. A036437.

Programs

Maple

T:= proc(n) option remember; local f, g;
      if n=0 then 1
    else f:= z-> add([T(n-1)][i]*z^(i-1), i=1..nops([T(n-1)]));
         g:= expand(1 +z*(f(z)^3/6 +f(z^2)*f(z)/2 +f(z^3)/3));
         seq(coeff(g, z, i), i=0..degree(g, z))
      fi
    end:
seq(T(n), n=0..5); # Alois P. Heinz, Sep 26 2011

Mathematica

T[n_] := T[n] = Module[{f, g}, If[n == 0, {1}, f[z_] = Sum[T[n-1][[i]]*z^(i-1), {i, 1, Length[T[n-1]]}]; g = Expand[1+z*(f[z]^3/6+f[z^2]*f[z]/2+f[z^3]/3)]; Table[Coefficient [g, z, i], {i, 0, Exponent[g, z]}]]]; Table[T[n], {n, 0, 5}] // Flatten (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)

Formula

T_{i+1}(z) = 1 +z*(T_i(z)^3/6 +T_i(z^2)*T_i(z)/2 +T_i(z^3)/3); T_0(z) = 1.