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A190174
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Number of vertices of even degree in the rooted tree with Matula-Goebel number n.
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1
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1, 0, 1, 1, 2, 2, 0, 0, 3, 3, 3, 1, 1, 1, 4, 1, 1, 2, 1, 2, 2, 4, 2, 2, 5, 2, 3, 0, 2, 3, 4, 0, 5, 2, 3, 3, 2, 2, 3, 3, 2, 1, 0, 3, 4, 3, 3, 1, 1, 4, 3, 1, 0, 4, 6, 1, 3, 3, 2, 4, 3, 5, 2, 1, 4, 4, 2, 1, 4, 2, 3, 2, 1, 3, 5, 1, 4, 2, 3, 2, 5, 3, 3, 2, 4, 1
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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.
The degree sequences of the rooted trees with Matula-Goebel number n are given in A182907.
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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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For a graph with degree sequence a,b,c,..., define the degree sequence polynomial to be x^a + x^b + x^c + ... . The degree sequence polynomial g(n)=g(n,x) of the rooted tree with Matula-Goebel number n can be obtained recursively in the following way: g(1)=1; if n=p(t) (=the t-th prime), then g(n)=g(t)+x^G(t)*(x-1)+x; if n=rs (r,s>=2), then g(n)=g(r)+g(s)-x^G(r)-x^G(s)+x^G(n); G(m) is the number of prime divisors of m counted with multiplicities. Clearly, a(n)=(1/2)*(g(n,1) + 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 and the vertex degrees are 1,1,2,2;
a(7)=0 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y having vertices of degree 1,1,1,3.
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MAPLE
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with(numtheory): g := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 1 elif bigomega(n) = 1 then sort(expand(g(pi(n))+x^bigomega(pi(n))*(x-1)+x)) else sort(expand(g(r(n))+g(s(n))-x^bigomega(r(n))-x^bigomega(s(n))+x^bigomega(n))) end if end proc: a := proc (n) options operator, arrow: (1/2)*subs(x = 1, g(n))+(1/2)*subs(x = -1, g(n)) end proc: seq(a(n), n = 1 .. 110);
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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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