

A184168


Sum of escape distances of all vertices in the rooted tree having MatulaGoebel 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
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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 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.


REFERENCES

F. Goebel, On a 11correspondence 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 YeongNan 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..110.
E. Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288, 2011
Index entries for sequences related to MatulaGoebel numbers


FORMULA

In A184167 one can find the generating polynomial P(n)=P(n,x) of the vertices of the rooted tree having MatulaGoebel 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 MatulaGoebel 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: A335320 A182182 A158823 * A122913 A289444 A069115
Adjacent sequences: A184165 A184166 A184167 * A184169 A184170 A184171


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Oct 23 2011


STATUS

approved



