A127535 Triangle read by rows: T(n,k) is the number of even trees with 2n edges and jump-length equal to k (0<=k<=n-1).

1, 2, 1, 4, 6, 2, 8, 22, 20, 5, 16, 66, 107, 70, 14, 32, 178, 428, 496, 252, 42, 64, 450, 1449, 2498, 2235, 924, 132, 128, 1090, 4410, 10234, 13662, 9878, 3432, 429, 256, 2562, 12479, 36558, 66107, 71370, 43043, 12870, 1430, 512, 5890, 33512, 118588

Offset: 1

Author

Emeric Deutsch, Jan 19 2007

Keywords

• nonn
• tabl

Comments

An even tree is an ordered tree in which each vertex has an even outdegree. In the preorder traversal of an ordered tree, any transition from a node at a deeper level to a node on a strictly higher level is called a jump; the positive difference of the levels is called the jump distance; the sum of the jump distances in a given ordered tree is called the jump-length.

The Krandick reference considers jumps and jump-length only in full binary trees.

Examples

			Triangle starts:
1;
2,1;
4,6,2;
8,22,20,5;
16,66,107,70,14;
		

Links

• W. Krandick, Trees and jumps and real roots, J. Computational and Applied Math., 162, 2004, 51-55.

Crossrefs

Cf. A001764, A000079, A000108, A127536, A127529, A127532.

Programs

Maple

eq:=(2*t-1-t^2+2*z-t*z)*G^3-(2+2*t*z-2*t-5*z)*G^2+(4*z-t*z-1)*G+z: g:=RootOf(eq,G): gser:=simplify(series(g,z=0,14)): for n from 1 to 11 do P[n]:=sort(expand(coeff(gser,z,n))) od: for n from 1 to 11 do seq(coeff(P[n],t,j),j=0..n-1) od; # yields sequence in triangular form

Formula

G.f.: G=G(t,z) is given by (2t-1-t^2+2z-tz)G^3-(2+2tz-2t-5z)G^2+(4z-tz-1)G+z=0.

Sum of terms in row n = C(3n,n)/(2n+1) (A001764).

T(n,0)=2^(n-1) (A000079).

T(n+1,n)=C(2n,n)/(n+1) (A000108, the Catalan numbers).

Sum(k*T(n,k),0<=k<=n-1)=A127536(n).