A127536 Sum of jump-lengths of all even trees with 2n edges.

0, 1, 10, 77, 546, 3740, 25194, 168245, 1118260, 7413705, 49085400, 324794316, 2148789800, 14217578856, 94096891658, 622997471685, 4126520887720, 27345271410275, 181295437422330, 1202538435463365, 7980245606038650

Offset

1,3

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.

Links

Table of n, a(n) for n=1..21.

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

Formula

a(n) = (n-1)(2n-1)C(3n,n)/[3(n+1)/(2n+1)].

a(n) = Sum_{k=0..n-1} k*A127535(n,k).

D-finite with recurrence 2*(n-2)*(2*n+1)*(2*n-3)*(n+1)*a(n) -3*(n-1)*(3*n-1)*(2*n-1)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Jul 26 2022

Maple

seq((n-1)*(2*n-1)*binomial(3*n, n)/3/(n+1)/(2*n+1), n=1..25);

Crossrefs

Cf. A127535, A127533.

Sequence in context: A244720 A081678 A081182 * A016201 A080618 A298270

Adjacent sequences: A127533 A127534 A127535 * A127537 A127538 A127539

Keyword

nonn

Author

Emeric Deutsch, Jan 19 2007

Status

approved