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A206491
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Irregular triangle read by rows: T(n,k) is the number of root subtrees with k nodes in the rooted tree having Matula-Goebel number n.
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5
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1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 2, 3, 2, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 3, 4, 3, 1, 1, 1, 2, 2, 1, 1, 2, 3, 3, 1, 1, 2, 3, 3, 2, 1, 1, 4, 6, 4, 1, 1, 1, 1, 2, 1, 1, 3, 5, 5, 3, 1, 1, 1, 3, 3, 1, 1, 3, 4, 4, 3, 1, 1, 2, 4, 4, 3, 1, 1, 2, 2, 2, 2, 1, 1, 1, 2, 3, 2, 1, 1, 4, 7, 7, 4, 1
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OFFSET
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1,8
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COMMENTS
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A root subtree of a rooted tree G is a subtree of G containing the root.
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.
Number of entries in row n = A061775(n).
Sum of entries in row n = A184160(n).
For the number of all subtrees of a given size, see A212620.
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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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EXAMPLE
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Row 7 is 1,1,2,1 because the rooted tree with Matula-Goebel number 7 is Y; its five root subtrees have 1, 2, 3, 3, and 4 nodes.
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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: R := proc (n, k) 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 and k = 1 then 1 elif n = 1 and 1 < k then 0 elif bigomega(n) = 1 and k = 1 then 1 elif bigomega(n) = 1 then R(pi(n), k-1) else add(R(r(n), j)*R(s(n), k+1-j), j = 1 .. k) end if end proc: for n to 40 do seq(R(n, k), k = 1 .. V(n)) end do; # yields sequence in triangular form
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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