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A191400
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Number of nonroot vertices of degree 2 in the rooted tree having Matula-Goebel number n.
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0
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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, 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, 3, 2, 2, 2, 1, 1, 5, 0, 3, 2, 3, 2, 4, 2, 3, 1, 3, 0
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OFFSET
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1,5
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COMMENTS
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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.
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REFERENCES
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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.
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LINKS
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FORMULA
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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).
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EXAMPLE
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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.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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