A127536 - OEIS

%I #11 Jul 26 2022 11:08:27

%S 0,1,10,77,546,3740,25194,168245,1118260,7413705,49085400,324794316,

%T 2148789800,14217578856,94096891658,622997471685,4126520887720,

%U 27345271410275,181295437422330,1202538435463365,7980245606038650

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

%C 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.

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

%H W. Krandick, <a href="https://doi.org/10.1016/j.cam.2003.08.018">Trees and jumps and real roots</a>, J. Computational and Applied Math., 162, 2004, 51-55.

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

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

%F 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

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

%Y Cf. A127535, A127533.

%K nonn

%O 1,3

%A _Emeric Deutsch_, Jan 19 2007