%I
%S 1,2,2,12,12,8,8,288,288,288,144,144,144,34560,64,144,10368,64,13824,
%T 13824,34560,10368,3456,24883200,10368,2985984,5184,13824,4976640,
%U 34560,1024,24883200,13824,8294400,746496,3456,3456,4976640,1327104,10368,1492992,5184,8294400,7166361600
%N The multiplicative Wiener index of the rooted tree with MatulaGoebel number n.
%C The multiplicative Wiener index of a connected graph is the product of the distances between all unordered pairs of vertices in the graph.
%C The MatulaGoebel number of a rooted tree is defined in the following recursive manner: to the onevertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the tth prime number, where t is the MatulaGoebel 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 MatulaGoebel numbers of the m branches of T.
%D F. Goebel, On a 11correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
%D I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
%D I. Gutman, W. Linert, I. Lukovits, and Z. Tomovic, The multiplicative version of the Wiener index, J. Chem. Inf. Comput. Sci., 40, 2000, 113116.
%D I. Gutman, W. Linert, I. Lukovits, and Z. Tomovic, On the multiplicative Wiener index and its possible chemical applications, Monatshefte f. Chemie, 131, 2000, 421427.
%D I. Gutman and YeongNan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
%D D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%H E. Deutsch, <a href="http://arxiv.org/abs/1111.4288">Tree statistics from Matula numbers</a>, arXiv preprint arXiv:1111.4288, 2011
%H <a href="/index/Mat#matula">Index entries for sequences related to MatulaGoebel numbers</a>
%F a(n)=Product(k^c(k), k=1..d), where d is the diameter of the rooted tree with MatulaGoebel number n, and c(k) is the number of pairs of nodes at distance k (all these data are contained in the Wiener polynomial; see A196059). The Maple program is based on the above.
%e a(7)=8 because the rooted tree with MatulaGoebel number 7 is the rooted tree Y with distances 1,1,1,2,2,2; product of distances is 8.
%e a(2^m) = 2^[m(m1)/2] because the rooted tree with MatulaGoebel number 2^m is a star with m edges and we have m distances 1 and m(m1)/2 distances 2.
%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: product(k^coeff(W(n), x, k), k = 1 .. degree(W(n))) end proc: seq(a(n), n = 2 .. 45);
%Y A196059
%K nonn
%O 2,2
%A _Emeric Deutsch_, Sep 30 2011
