OFFSET
0,2
COMMENTS
In the preorder traversal of a binary tree, any transition from a node at a deeper level to a node on a strictly higher level is called a jump.
The Krandick reference considers the statistic "number of jumps" for full binary trees.
Row 0 has one term, row n (n >= 1) has ceiling(n/2) terms.
LINKS
W. Krandick, Trees and jumps and real roots, J. Computational and Applied Math., 162, 2004, 51-55.
FORMULA
EXAMPLE
Triangle starts:
1;
2;
5;
12, 2;
29, 13;
70, 60, 2;
169, 235, 25;
MAPLE
G:= (-z^2-2*z+z^2*t+1-sqrt(z^4+4*z^3-2*z^4*t+2*z^2-4*z^3*t-4*z+z^4*t^2-2*z^2*t+1))/2/t/z^2: Gser:=simplify(series(G, z=0, 17)): for n from 1 to 14 do P[n]:=sort(coeff(Gser, z, n)) od: 1; for n from 0 to 14 do seq(coeff(P[n], t, j), j=0..ceil(n/2)-1) od; # yields sequence in triangular form
MATHEMATICA
n = 13; g[t_, z_] := (-z^2 - 2z + z^2*t + 1 - Sqrt[z^4 + 4z^3 - 2z^4*t + 2z^2 - 4z^3*t - 4z + z^4*t^2 - 2z^2*t + 1])/2/t/z^2; Flatten[ CoefficientList[#1, t] & /@ CoefficientList[Simplify[Series[g[t, z], {z, 0, n}]], z]] (* Jean-François Alcover, Jul 22 2011, after g.f. *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Jan 18 2007
STATUS
approved