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

1,6

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

Formula

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

Example

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

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 *)

Crossrefs

Sequence in context: A269300 A182215 A036443 * A053306 A108422 A276523

Adjacent sequences: A036434 A036435 A036436 * A036438 A036439 A036440

Keyword

nonn,easy,tabf

Author

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

Status

approved