

A352288


Total cophenetic index of the rooted tree with MatulaGoebel number n.


2



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
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OFFSET

1,17


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)


LINKS

Table of n, a(n) for n=1..87.
Arnau Mir, Francesc Rosselló, and Lucía Rotger, A New Balance Index for Phylogenetic Trees, arXiv:1202.1223 [qbio.PE], 2012.
Kevin Ryde, PARI/GP Code
Index entries for sequences related to MatulaGoebel numbers


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.


EXAMPLE

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


PROG

(PARI) See links.


CROSSREFS

Cf. A348959 (terminal Wiener), A196048 (external length), A109129 (childless vertices).
Cf. A325663 (indices of 0's), A352289 (max by leaves).
Sequence in context: A078442 A175663 A240672 * A243016 A284975 A219202
Adjacent sequences: A352285 A352286 A352287 * A352289 A352290 A352291


KEYWORD

nonn


AUTHOR

Kevin Ryde, Mar 16 2022


STATUS

approved



