%N The hyper-Wiener index of the rooted tree with Matula-Goebel number n.
%C 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.
%D G. G. Cash, Relationship between the Hosoya polynomial and the hyper-Wiener index, Appl. Math. Letters, 15, 2002, 893-895.
%D E. Deutsch, Tree statistics from Matula numbers, Arxiv preprint arXiv:1111.4288, 2011
%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. J. Klein, I. Lukovits and I. Gutman, On the definition of the hyper-Wiener index for cycle-containing structures, J. Chem. Inf. Comput. Sci., 35, 1995, 50-52.
%D D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%F a(n)=W'(n,1)+(1/2)W"(n,1), where W(n,x) is the Wiener polynomial (called also 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.
%e 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.
%e 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).
%p 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);
%A _Emeric Deutsch_, Sep 30 2011