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A196060
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The hyper-Wiener index of the rooted tree with Matula-Goebel number n.
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3
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0, 1, 5, 5, 15, 15, 12, 12, 35, 35, 35, 28, 28, 28, 70, 22, 28, 54, 22, 58, 58, 70, 54, 44, 126, 54, 90, 47, 58, 99, 70, 35, 126, 58, 108, 76, 44, 44, 99, 84, 54, 83, 47, 108, 150, 90, 99, 63, 91, 165, 108, 83, 35, 118, 210, 69, 84, 99, 58, 131, 76, 126, 129, 51, 170, 170, 44, 91, 150, 143, 84, 101, 83, 76, 231
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OFFSET
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1,3
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COMMENTS
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The hyper-Wiener index of a connected graph is (1/2)*Sum [d(i,j)+d(i,j)^2], where d(i,j) is the distance between the vertices i and j and summation is over all unordered pairs of vertices (i,j).
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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a(n) = W'(n,1) + (1/2)W"(n,1), where W(n,x) is the Wiener polynomial (also called Hosoya polynomial) of the rooted tree with Matula-Goebel index n. W(n)=W(n,x) is obtained recursively in A196059. The Maple program is based on the above.
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EXAMPLE
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a(7)=12 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y; the distances are 1,1,1,2,2,2; sum of distances = 9; sum of squared distances = 15; (9+15)/2=12.
a(2^m) = m(3m-1)/2 because the rooted tree with Matula-Goebel number 2^m is a star with m edges and we have m distances 1 and m(m-1)/2 distances 2; sum of the distances = m^2; sum of the squared distances = 2m^2 - m; hyper-Wiener index is (1/2)(3m^2 - m).
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MAPLE
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with(numtheory): W := proc (n) local r, s, R: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: R := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then sort(expand(x*R(pi(n))+x)) else sort(expand(R(r(n))+R(s(n)))) end if end proc: if n = 1 then 0 elif bigomega(n) = 1 then sort(expand(W(pi(n))+x*R(pi(n))+x)) else sort(expand(W(r(n))+W(s(n))+R(r(n))*R(s(n)))) end if end proc: a := proc (n) options operator, arrow: subs(x = 1, diff(W(n), x)+(1/2)*(diff(W(n), `$`(x, 2)))) end proc: seq(a(n), n = 1 .. 75);
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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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