A127535 -id:A127535 - cp's OEIS frontend

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

Emeric Deutsch, Jan 19 2007

nonn

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.

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

Cf. A127535, A127533.

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

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

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

A127534 Number of jumps in all even trees with 2n edges.

Original entry on oeis.org

0, 1, 9, 65, 442, 2940, 19380, 127281, 834900, 5476185, 35937525, 236030652, 1551652424, 10210456360, 67254204696, 443410005585, 2926078447656, 19325957314755, 127746785056275, 845069382939705, 5594334252541650

Offset: 1

Emeric Deutsch, Jan 19 2007

nonn

The Krandick reference considers jumps in full binary trees.

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

Cf. A127535, A127536.

seq((n-1)*(4*n-3)*binomial(3*n,n)/3/(2*n+1)/(3*n-1),n=1..24);

Table[((n-1)(4n-3)Binomial[3n,n])/(3(2n+1)(3n-1)),{n,30}] (* Harvey P. Dale, Sep 29 2013 *)

a(n)=(n-1)(4n-3)C(3n,n)/[3(2n+1)(3n-1)].

D-finite with recurrence 8*n*(2*n+1)*a(n) -2*(136*n-69)*(n-1)*a(n-1) +5*(263*n^2-893*n+750)*a(n-2) -156*(3*n-8)*(3*n-10)*a(n-3)=0. - R. J. Mathar, Jul 22 2022

cp's OEIS Frontend

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

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