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 A184157 The sum of the even distances in the rooted tree with Matula-Goebel number n. 1
 0, 0, 2, 2, 4, 4, 6, 6, 10, 10, 10, 8, 8, 8, 16, 12, 8, 14, 12, 18, 18, 16, 14, 14, 28, 14, 24, 12, 18, 24, 16, 20, 28, 18, 24, 20, 14, 14, 24, 28, 14, 22, 12, 24, 34, 24, 24, 22, 30, 40, 24, 22, 20, 30, 40, 18, 28, 24, 18, 34, 20, 28, 36, 30, 36, 36, 14, 30, 34, 32, 28, 28, 22, 20, 50, 18, 42, 32, 24, 40 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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. a(n) + A184158(n) = A196051(n) (= the Wiener index of the rooted tree with Matula-Goebel number n). REFERENCES F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143. I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142. I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22. 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. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273. LINKS E. Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288, 2011 FORMULA 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. EXAMPLE a(7)=6 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y with 3 distances equal to 2. MAPLE 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); CROSSREFS Cf. A184158, A196051 Sequence in context: A328422 A029009 A023023 * A008643 A008644 A008645 Adjacent sequences:  A184154 A184155 A184156 * A184158 A184159 A184160 KEYWORD nonn AUTHOR Emeric Deutsch, Oct 15 2011 STATUS approved

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Last modified August 11 09:30 EDT 2020. Contains 336423 sequences. (Running on oeis4.)