%I
%S 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,
%T 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,
%U 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
%N Sum of escape distances of all vertices in the rooted tree having MatulaGoebel number n.
%C 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).
%C 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.
%D F. Goebel, On a 11correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
%D I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
%D I. Gutman and YeongNan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
%D D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%H E. Deutsch, <a href="http://arxiv.org/abs/1111.4288">Tree statistics from Matula numbers</a>, arXiv preprint arXiv:1111.4288, 2011
%H <a href="/index/Mat#matula">Index entries for sequences related to MatulaGoebel numbers</a>
%F 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.
%e 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.
%p 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);
%Y Cf. A184167
%K nonn
%O 1,3
%A _Emeric Deutsch_, Oct 23 2011
