A036437 Triangle of coefficients of generating function of ternary rooted trees of height exactly n.

1, 1, 1, 1, 1, 2, 4, 4, 5, 4, 4, 3, 2, 1, 1, 1, 3, 8, 15, 27, 43, 67, 97, 136, 183, 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, 1, 4, 13, 32, 74, 155, 316, 612, 1160, 2126, 3829, 6737

Offset: 1

Author

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

Keywords

• nonn
• easy
• tabf

Examples

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

Links

• Alois P. Heinz, Rows n = 1..8, flattened
• A. T. Balaban, J. W. Kennedy and L. V. Quintas, The number of alkanes having n carbons and a longest chain of length d, J. Chem. Education, 65 (1988), 304-313.
• Index entries for sequences related to rooted trees

Programs

Maple

df:= (t, l)-> zip((x,y)->x-y, t, l, 0):
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(df([T(n)], [T(n-1)])[n+1..-1][], n=1..5); # Alois P. Heinz, Sep 26 2011

Mathematica

df[t_, l_] := Plus @@ PadRight[{t, -l}]; 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[df[T[n], T[n-1]][[n+1 ;; -1]], {n, 1, 5}] // Flatten (* Jean-François Alcover, Jan 30 2014, after Alois P. Heinz *)

Formula

T_{n}(z) - T_{n-1}(z) (see A036370).