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A206493
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Product, over all vertices v of the rooted tree with Matula-Goebel number n, of the number of vertices in the subtree with root v.
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2
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1, 2, 6, 3, 24, 8, 12, 4, 20, 30, 120, 10, 40, 15, 72, 5, 60, 24, 20, 36, 36, 144, 120, 12, 252, 48, 56, 18, 180, 84, 720, 6, 336, 72, 126, 28, 60, 24, 112, 42, 240, 42, 90, 168, 192, 140, 504, 14, 63, 288, 168, 56, 30, 64, 1152, 21, 56, 210, 360, 96, 168, 840, 96, 7, 384
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OFFSET
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1,2
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COMMENTS
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a(n) is called tree factorial. See, for example, the Brouder reference.
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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LINKS
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FORMULA
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Denote by V(k) the number of vertices of the rooted tree with Matula-Goebel number k. If n is the m-th prime, then a(n) = a(m)*V(n); if n=rs, r,s>=2, then a(n) = a(r)a(s)V(n)/{V(r)V(s)}. The Maple program is based on these recurrence relations.
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EXAMPLE
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a(7)=12 because the rooted tree with Matula-Goebel number 7 is Y; denoting the vertices in preorder by a,b,c, and d, the number of vertices of the subtrees having these roots are 4, 3, 1, and 1, respectively. a(11)=120 because the rooted tree with Matula-Goebel number 11 is the path tree on 5 vertices; the subtrees have 5,4,3,2,1 vertices.
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MAPLE
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with(numtheory): V := 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 1+V(pi(n)) else V(r(n))+V(s(n))-1 end if end proc: H := 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 V(n)*H(pi(n)) else H(r(n))*H(s(n))*V(n)/(V(r(n))*V(s(n))) end if end proc: seq(H(n), n = 1 .. 100);
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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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