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A212624
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Number of vertices in all independent vertex subsets of the rooted tree with Matula-Goebel number n.
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10
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1, 2, 5, 5, 10, 10, 13, 13, 20, 20, 20, 23, 23, 23, 38, 33, 23, 41, 33, 45, 45, 38, 41, 55, 71, 41, 74, 48, 45, 78, 38, 81, 71, 45, 82, 92, 55, 55, 78, 105, 41, 85, 48, 82, 137, 74, 78, 131, 98, 146, 82, 85, 81, 155, 130, 108, 105, 78, 45, 173, 92, 71, 153, 193, 141, 141, 55, 98, 137, 157, 105, 212
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OFFSET
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1,2
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COMMENTS
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A vertex subset in a tree is said to be independent if no pair of vertices is connected by an edge. The empty set is considered to be independent.
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 Y-N. 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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In A212623 one finds the generating polynomial P(n,x) with respect to the number of vertices of the independent vertex subsets of the rooted tree with Matula-Goebel number n. We have a(n) = subs(x=1, (d/dx)P(n,x)).
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EXAMPLE
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a(5)=10 because the rooted tree with Matula-Goebel number 5 is the path tree R - A - B - C with independent vertex subsets: {}, {R}, {A}, {B}, {C}, {R,B}, {R,C}, {A,C}. The total number of vertices is 10.
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MAPLE
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with(numtheory): A := 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 [x, 1] elif bigomega(n) = 1 then [expand(x*A(pi(n))[2]), expand(A(pi(n))[1])+A(pi(n))[2]] else [sort(expand(A(r(n))[1]*A(s(n))[1]/x)), sort(expand(A(r(n))[2]*A(s(n))[2]))] end if end proc: P := proc (n) options operator, arrow: sort(A(n)[1]+A(n)[2]) end proc: a := proc (n) options operator, arrow: subs(x = 1, diff(P(n), x)) end proc: seq(a(n), n = 1 .. 100);
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CROSSREFS
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Cf. A212618, A212619, A212620, A212621, A212622, A212623, A212625, A212626, A212627, A212628, A212629, A212630, A212631, A212632.
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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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