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 A352288 Total cophenetic index of the rooted tree with Matula-Goebel 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 (list; graph; refs; listen; history; text; internal format)
 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 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 FORMULA a(n) = Sum_{i=1..k} a(primepi(p[i])) + binomial(C(p[i]),2), where n = p*...*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

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Last modified October 5 05:10 EDT 2022. Contains 357252 sequences. (Running on oeis4.)