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%I #10 Mar 07 2017 11:26:49
%S 0,1,7,7,28,28,30,30,84,84,84,94,94,94,210,104,94,247,104,243,243,210,
%T 247,283,462,247,579,278,243,565,210,320,462,243,547,681,283,283,565,
%U 667,247,661,278,547,1216,579,565,793,644,1174,547,661,320,1506,924
%N The overall hyper-Wiener index of the rooted tree having Matula-Goebel number n.
%C The overall hyper-Wiener index of any connected graph G is defined as the sum of the hyper-Wiener indices of all the subgraphs of G. 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).
%C 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.
%C The Maple program yields a(n) by using the command OHW(n) for n<=3000 (adjustable).
%D F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
%D I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
%D I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
%D D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%D D. Bonchev, The overall Wiener index - a new tool for characterization of molecular topology, J. Chem. Inf. Comput. Sci., 2001, 41, 582-592.
%D X. H. Li and J. J. Lin, The overall hyper-Wiener index, J. Math. Chemistry, 33, 2003, 81-89.
%H E. Deutsch, <a href="http://arxiv.org/abs/1111.4288"> Rooted tree statistics from Matula numbers</a>, arXiv:1111.4288.
%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A198339(n) gives the sequence of the Matula-Goebel numbers of all the subtrees of the rooted tree with Matula-Goebel number n. A196060(k) is the hyper-Wiener index of the rooted tree with Matula-Goebel number k.
%e a(4)=7 because the rooted tree with Matula-Goebel number 4 is V; each of the 3 one-vertex subtrees has hyper-Wiener index 0, each of the 2 one-edge subtrees has hyper-Wiener index 1, and the tree V itself has hyper-Wiener index 5; 0+0+0+1+1+5=7.
%p m2union := proc (x, y) sort([op(x), op(y)]) end proc: with(numtheory); MRST := 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, seq(ithprime(mrst[pi(n)][i]), i = 1 .. nops(mrst[pi(n)]))] else [seq(seq(mrst[r(n)][i]*mrst[s(n)][j], i = 1 .. nops(mrst[r(n)])), j = 1 .. nops(mrst[s(n)]))] end if end proc: MNRST := 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 [] elif bigomega(n) = 1 then m2union(mrst[pi(n)], mnrst[pi(n)]) else m2union(mnrst[r(n)], mnrst[s(n)]) end if end proc: MST := proc (n) m2union(mrst[n], mnrst[n]) end proc: for n to 3000 do mrst[n] := MRST(n): mnrst[n] := MNRST(n): mst[n] := MST(n) end do: 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: HW := proc (n) options operator, arrow: subs(x = 1, diff(W(n), x)+(1/2)*(diff(W(n), `$`(x, 2)))) end proc: OHW := proc (n) options operator, arrow: add(HW(MST(n)[j]), j = 1 .. nops(MST(n))) end proc: seq(OHW(n), n = 1 .. 60);
%Y Cf. A196060, A198339.
%K nonn
%O 1,3
%A _Emeric Deutsch_, Dec 04 2011
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