A036370 - OEIS

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

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OFFSET

0,12

LINKS

Alois P. Heinz, Rows n = 0..8, flattened

Index entries for sequences related to rooted trees

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.

EXAMPLE

1, 1;

1, 1, 1, 1, 1;

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

...

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))

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 *)

CROSSREFS

Cf. A036437.

Sequence in context: A319715 A088807 A036371 * A005208 A110007 A327715

Adjacent sequences: A036367 A036368 A036369 * A036371 A036372 A036373

KEYWORD

nonn,easy,tabf

AUTHOR

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

STATUS

approved