A352288 Total cophenetic index of the rooted tree with Matula-Goebel number n.

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 3, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 2, 1, 0, 3, 3, 1, 0, 2, 1, 4, 0, 0, 1, 1, 0, 2, 0, 2, 1, 6, 0, 0, 1, 3, 1, 3, 0, 3, 0, 1, 0, 1, 0, 6, 2, 1, 1, 3, 0, 4, 3, 0, 3, 1, 1, 1, 0, 0, 2, 2, 1, 2, 4, 1

Offset: 1

Author

Kevin Ryde, Mar 16 2022

Keywords

• nonn

Comments

Mir, Rosselló, and Rotger, define the cophenetic value of a pair of childless vertices as the depth (distance down from the root) of their deepest common ancestor, and they then define the total cophenetic index of a tree as the sum of the cophenetic values over all such pairs.

a(n) = 0 iff n is in A325663, being rooted stars with any arm lengths, since the root (depth 0) is the common ancestor of every childless pair.

An identity relating the childless terminal Wiener index TW(n) = A348959(n) can be constructed by noting it measures distances from a pair of childless vertices to their common ancestor, and the cophenetic values measure from that ancestor up to the root. So 2*a(n) + TW(n) is total depths Ext(n) = A196048(n) of the childless vertices, repeated by childless vertices C(n) = A109129(n) except itself, so that 2*a(n) + TW(n) = Ext(n)*(C(n) - 1)

Examples

			For n=111, the tree and its childless pairs and deepest common ancestors are
  root  R         pair  ancestor depth
       / \         G,D     A       1
      A   B        G,E     A       1
     /|\   \       D,E     A       1
    C D E   F     any,F    R       0
    |                             ---
    G                 total a(n) = 3
		

Links

• Arnau Mir, Francesc Rosselló, and Lucía Rotger, A New Balance Index for Phylogenetic Trees, arXiv:1202.1223 [q-bio.PE], 2012.
• Kevin Ryde, PARI/GP Code
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A348959 (terminal Wiener), A196048 (external length), A109129 (childless vertices).

Cf. A325663 (indices of 0's), A352289 (max by leaves).

Programs

PARI

\\ See links.

Formula

a(n) = Sum_{i=1..k} a(primepi(p[i])) + binomial(C(p[i]),2), where n = p[1]*...*p[k] is the prime factorization of n with multiplicity (A027746), and C(n) = A109129(n) is the number of childless vertices.