

A206499


The sum of the distances between all unordered pairs of branch vertices in the rooted tree with MatulaGoebel number n. A branch vertex is a vertex of degree >=3.


4



0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 4, 0, 0, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,49


COMMENTS

The MatulaGoebel number of a rooted tree can be defined in the following recursive manner: to the onevertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the tth prime number, where t is the MatulaGoebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the MatulaGoebel numbers of the m branches of T.
The A. Ilic and M. Ilic reference considers the statistic: the sum of the distances between all unordered pairs of vertices of degree k (see A212618, A212619).


REFERENCES

F. Goebel, On a 11 correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
I. Gutman and YN. Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.


LINKS

Table of n, a(n) for n=1..101.
E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288.
A. Ilic and M. Ilic, Generalizations of Wiener polarity index and terminal Wiener index, arXiv:11106.2986.
Index entries for sequences related to MatulaGoebel numbers


FORMULA

Let bigomega(n) denote the number of prime divisors of n, counted with multiplicities. Let g(n)=g(n,x) be the generating polynomial of the branch vertices of the rooted tree with MatulaGoebel number n with respect to level. We have a(1) = 0; if n = p(t) (=the tth prime) and bigomega(t) is not 2, then a(n) = a(t); if n = p(t) (=the tth prime) and bigomega(t) = 2, then a(n) = a(t) + [dg(t)/dx]_{x=1}; if n = rs with r prime, bigomega(s) =/ 2, then a(n) = a(r) + a(s) + [d[g(r)g(s)]/dx]_{x=1}; if n=rs with r prime, bigomega(s)=2, then a(n)=a(r)+a(s)+ [d[g(r)g(s)]/dx]_{x=1} + [dg(r)/dx]_{x=1} + [dg(s)/dx]_{x=1}.


EXAMPLE

a(28)=1 because the rooted tree with MatulaGoebel number 28 is the rooted tree obtained by joining the trees I, I, and Y at their roots; it has 2 branch vertices and the distance between them is 1. a(49)=2 because the rooted tree with MatulaGoebel number 49 is the rooted tree obtained by joining two copies of Y at their roots; it has 2 branch vertices and the distance between them is 2.


MAPLE

with(numtheory): g := proc (n) local r, s: r := proc(n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 0 elif bigomega(n) = 1 and bigomega(pi(n)) <> 2 then sort(expand(x*g(pi(n)))) elif bigomega(n) = 1 and bigomega(pi(n)) = 2 then sort(expand(x+x*g(pi(n)))) elif bigomega(r(n))+bigomega(s(n)) = 2 then sort(expand(g(r(n))subs(x = 0, g(r(n)))+g(s(n))subs(x = 0, g(s(n))))) else sort(expand(g(r(n))subs(x = 0, g(r(n)))+g(s(n))subs(x = 0, g(s(n)))+1)) end if end proc: a := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 0 elif bigomega(n) = 1 and bigomega(pi(n)) <> 2 then a(pi(n)) elif bigomega(n) = 1 and bigomega(pi(n)) = 2 then a(pi(n))+subs(x = 1, diff(g(pi(n)), x)) elif bigomega(s(n)) = 2 then a(r(n))+a(s(n))+subs(x = 1, diff((1+g(r(n)))*(1+g(s(n))), x)) else a(r(n))+a(s(n))+subs(x = 1, diff(g(r(n))*g(s(n)), x)) end if end proc: seq(a(n), n = 1 .. 120);


CROSSREFS

Cf. A212618, A212619.
Sequence in context: A176891 A219486 A284574 * A277885 A109527 A186715
Adjacent sequences: A206496 A206497 A206498 * A206500 A206501 A206502


KEYWORD

nonn


AUTHOR

Emeric Deutsch, May 22 2012


STATUS

approved



