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A036370 Triangle of coefficients of generating function of ternary rooted trees of height at most n. 19

%I

%S 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,

%T 70,99,137,184,239,300,369,432,498,551,594,614,624,601,570,514,453,

%U 378,312,238,181,128,89,56,37,20,12,6,3,1,1

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

%H Alois P. Heinz, <a href="/A036370/b036370.txt">Rows n = 0..8, flattened</a>

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%F 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.

%e 1;

%e 1, 1;

%e 1, 1, 1, 1, 1;

%e 1, 1, 1, 2, 3, 4, 4, 5, 4, 4, 3, 2, 1, 1;

%e ...

%p T:= proc(n) option remember; local f, g;

%p if n=0 then 1

%p else f:= z-> add([T(n-1)][i]*z^(i-1), i=1..nops([T(n-1)]));

%p g:= expand(1 +z*(f(z)^3/6 +f(z^2)*f(z)/2 +f(z^3)/3));

%p seq(coeff(g, z, i), i=0..degree(g, z))

%p fi

%p end:

%p seq(T(n), n=0..5); # _Alois P. Heinz_, Sep 26 2011

%t 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-Fran├žois Alcover_, Mar 10 2014, after _Alois P. Heinz_ *)

%Y Cf. A036437.

%K nonn,easy,tabf

%O 0,12

%A _N. J. A. Sloane_, Eric Rains (rains(AT)caltech.edu)

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Last modified August 2 06:04 EDT 2021. Contains 346409 sequences. (Running on oeis4.)