A036437 - OEIS

%I #28 Mar 22 2024 08:56:20

%S 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,

%T 369,432,498,551,594,614,624,601,570,514,453,378,312,238,181,128,89,

%U 56,37,20,12,6,3,1,1,1,4,13,32,74,155,316,612,1160,2126,3829,6737

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

%H Alois P. Heinz, <a href="/A036437/b036437.txt">Rows n = 1..8, flattened</a>

%H A. T. Balaban, J. W. Kennedy and L. V. Quintas, <a href="https://doi.org/10.1021/ed065p304">The number of alkanes having n carbons and a longest chain of length d</a>, J. Chem. Education, 65 (1988), 304-313.

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

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

%e 1;

%e 1, 1, 1;

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

%p df:= (t, l)-> zip((x,y)->x-y, t, l, 0):

%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(df([T(n)], [T(n-1)])[n+1..-1][], n=1..5); # _Alois P. Heinz_, Sep 26 2011

%t 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_ *)

%K nonn,easy,tabf

%O 1,6

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