%N The sum of the even distances in the rooted tree with Matula-Goebel number n.
%C The Matula-Goebel number of a rooted tree is 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 a(n) + A184158(n) = A196051(n) (= the Wiener index of the rooted tree with Matula-Goebel number n).
%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 O. Ivanciuc, T. Ivanciuc, D. J. Klein, W. A. Seitz, and A. T. Balaban, Wiener index extension by counting even/odd graph distances, J. Chem. Inf. Comput. Sci., 41, 2001, 536-549.
%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 Matula-Goebel numbers</a>
%F a(n) is the value at x=1 of the derivative of the even part of the Wiener polynomial W(n)=W(n,x) of the rooted tree with Matula number n. W(n) is obtained recursively in A196059. The Maple program is based on the above.
%e a(7)=6 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y with 3 distances equal to 2.
%p with(numtheory): WP := 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(WP(pi(n))+x*R(pi(n))+x)) else sort(expand(WP(r(n))+WP(s(n))+R(r(n))*R(s(n)))) end if end proc: a := proc (n) options operator, arrow: (1/2)*subs(x = 1, diff(WP(n), x))-(1/2)*subs(x = -1, diff(WP(n), x)) end proc: seq(a(n), n = 1 .. 80);
%Y Cf. A184158, A196051
%A _Emeric Deutsch_, Oct 15 2011