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%I #10 Mar 07 2017 06:50:32
%S 0,0,1,0,2,1,0,0,2,2,3,1,1,0,3,0,1,2,0,2,1,3,2,1,4,1,3,0,2,3,4,0,4,1,
%T 2,2,1,0,2,2,2,1,0,3,4,2,3,1,0,4,2,1,0,3,5,0,1,2,2,3,2,4,2,0,3,4,1,1,
%U 3,2,2,2,1,1,5,0,3,2,3,2,4,2,3,1,3,0
%N Number of nonroot vertices of degree 2 in the rooted tree having Matula-Goebel number n.
%C 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.
%D F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
%D I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
%D I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
%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"> Rooted tree statistics from Matula numbers</a>, arXiv:1111.4288
%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F Let g(n)=G(n,x) be the generating polynomial of the nonroot nodes of degree 2 of the rooted tree having Matula-Goebel number n, with respect to level. Then g(1)=g(2)=0; if n = p(t) (=the t-th prime) and t is prime, then g(n)=x+x*g(t); if n=p(t) (=the t-th prime) and t is not prime, then g(n)=x*g(t); if n=rs (r,s>=2), then g(n)=g(r)+g(s). Clearly, a(n)=G(n,1).
%e a(5)=2 because the rooted tree with Matula-Goebel number 5 is the path tree on 4 vertices. a(7)=0 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y, with no vertices of degree 2.
%K nonn
%O 1,5
%A _Emeric Deutsch_, Dec 10 2011
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