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A184168 Sum of escape distances of all vertices in the rooted tree having Matula-Goebel number n. 0
0, 1, 3, 1, 6, 2, 3, 1, 4, 4, 10, 2, 4, 2, 6, 1, 6, 3, 3, 4, 4, 7, 7, 2, 9, 3, 5, 2, 6, 5, 15, 1, 9, 4, 6, 3, 4, 2, 5, 4, 7, 3, 4, 7, 7, 5, 9, 2, 4, 7, 6, 3, 3, 4, 12, 2, 4, 5, 10, 5, 5, 11, 5, 1, 7, 8, 6, 4, 7, 5, 6, 3, 7, 3, 9, 2, 9, 4, 9, 4, 6, 5, 11, 3, 9, 3, 7, 7, 4, 6, 5, 5, 13, 7, 6, 2, 13, 3, 10, 7, 5, 5, 8, 3, 7, 2, 4, 4, 9, 10 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The escape distance of a vertex v in a rooted tree T is the distance from v to the nearest leaf of T that is a descendant of v. For the rooted tree ARBCDEF, rooted at R, the escape distance of B is 4 (the leaf A is not a descendant of B).

The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel 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 Matula-Goebel numbers of the m branches of T.

REFERENCES

F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.

I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.

I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.

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..110.

E. Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288, 2011

Index entries for sequences related to Matula-Goebel numbers

FORMULA

In A184167 one can find the generating polynomial P(n)=P(n,x) of the vertices of the rooted tree having Matula-Goebel number n, according to escape distance. a(n) is equal to dP/dx, evaluated at x=1.

EXAMPLE

a(7)=3 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y, having 4 vertices with escape distances 0,0,1, and 2.

MAPLE

with(numtheory): P := proc (n) local r, s, LLL: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: LLL := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+LLL(pi(n)) else min(LLL(r(n)), LLL(s(n))) end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then P(pi(n))+x^(1+LLL(pi(n))) else P(r(n))+P(s(n))-x^max(LLL(r(n)), LLL(s(n))) end if end proc: a := proc (n) options operator, arrow: subs(x = 1, diff(P(n), x)) end proc: seq(a(n), n = 1 .. 110);

CROSSREFS

Cf. A184167

Sequence in context: A010467 A182182 A158823 * A122913 A289444 A069115

Adjacent sequences:  A184165 A184166 A184167 * A184169 A184170 A184171

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Oct 23 2011

STATUS

approved

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Last modified January 18 13:09 EST 2019. Contains 319271 sequences. (Running on oeis4.)