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

%I A036437 #28 Mar 22 2024 08:56:20
%S A036437 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 A036437 369,432,498,551,594,614,624,601,570,514,453,378,312,238,181,128,89,
%U A036437 56,37,20,12,6,3,1,1,1,4,13,32,74,155,316,612,1160,2126,3829,6737
%N A036437 Triangle of coefficients of generating function of ternary rooted trees of height exactly n.
%H A036437 Alois P. Heinz, <a href="/A036437/b036437.txt">Rows n = 1..8, flattened</a>
%H A036437 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 A036437 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A036437 T_{n}(z) - T_{n-1}(z) (see A036370).
%e A036437 1;
%e A036437 1, 1, 1;
%e A036437 1, 2, 4, 4, 5, 4, 4, 3, 2, 1, 1;
%p A036437 df:= (t, l)-> zip((x,y)->x-y, t, l, 0):
%p A036437 T:= proc(n) option remember; local f, g;
%p A036437       if n=0 then 1
%p A036437     else f:= z-> add([T(n-1)][i]*z^(i-1), i=1..nops([T(n-1)]));
%p A036437          g:= expand(1 +z*(f(z)^3/6 +f(z^2)*f(z)/2 +f(z^3)/3));
%p A036437          seq(coeff(g, z, i), i=0..degree(g, z))
%p A036437       fi
%p A036437     end:
%p A036437 seq(df([T(n)], [T(n-1)])[n+1..-1][], n=1..5); # _Alois P. Heinz_, Sep 26 2011
%t A036437 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 A036437 nonn,easy,tabf
%O A036437 1,6
%A A036437 _N. J. A. Sloane_, Eric Rains (rains(AT)caltech.edu)