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

Author

Emeric Deutsch, Jan 19 2007

Keywords

• nonn

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

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

Crossrefs

Cf. A127535, A127533.

Programs

Maple

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

Mathematica

Table[(n - 1) (2 n - 1) Binomial[3 n, n]/3/(n + 1)/(2*n + 1), {n, 30}] (* Wesley Ivan Hurt, Aug 04 2025 *)

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