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 A212618 Sum of the distances between all unordered pairs of vertices of degree 2 in the rooted tree with Matula-Goebel number n. 15
 0, 0, 0, 0, 1, 1, 0, 0, 4, 4, 4, 0, 0, 0, 10, 0, 0, 2, 0, 1, 1, 10, 2, 0, 20, 2, 6, 0, 1, 6, 10, 0, 20, 1, 4, 2, 0, 0, 6, 1, 2, 0, 0, 4, 13, 6, 6, 0, 0, 14, 4, 0, 0, 6, 35, 0, 1, 6, 1, 6, 2, 20, 2, 0, 13, 13, 0, 0, 13, 1, 1, 2, 0, 2, 24, 0, 10, 3, 4, 1, 12, 6, 6, 0, 10, 0, 14, 4, 0, 13, 2, 2, 35, 13 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,9 COMMENTS 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. 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 Y-N. Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22. D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273. LINKS E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288. A. Ilic and M. Ilic, Generalizations of Wiener polarity index and terminal Wiener index, arXiv:11106.2986. FORMULA We give recurrence formulas for the more general case of vertices of degree k (k>=2). Let bigomega(n) denote the number of prime divisors of n, counted with multiplicities. Let g(n)=g(n,k,x) be the generating polynomial of the vertices of degree k of the rooted tree with Matula-Goebel number n with respect to level. We have a(1) = 0; if n = p(t) (=the t-th prime) and bigomega(t)=k-1 then a(n) = a(t)+[dg(t)/dx]_{x=1}; if n = p(t) (=the t-th prime) and bigomega(t)=k, then a(n) = a(t)-[dg(t)/dx]_{x=1};  if n = p(t) (=the t-th prime) and bigomega(t) =/ k and =/ k-1, then a(n)= a(t); if n = rs with r prime, s>=2, bigomega(s)=k-1, then a(n) = a(r) + a(s) + [d[g(r)g(s)]/dx]_{x=1} +[dg(r)/dx]_{x=1} +[dg(s)/dx]_{x=1}; if n = rs with r prime, s>=2, bigomega(s) =k, then a(n) = a(r) + a(s) + [d[g(r)g(s)]/dx]_{x=1} - [dg(r)/dx]_{x=1} - [dg(s)/dx]_{x=1}; if n = rs with r prime, s>=2, bigomega(s) =/ k-1 and =/ k, then a(n) = a(r) + a(s) + [d[g(r)g(s)]/dx]_{x=1}. EXAMPLE a(11)=4 because the rooted tree with Matula-Goebel number 11 is the path tree A - B - C - D - E; the vertices of degree 2 are B, C, and D; we have dist(B,C)+dist(B,D)+dist(C,D) = 1+2+1 = 4. a(987654321) = 68, as given by the Maple program; the reader can verify this on the rooted tree of Fig. 2 of the Deutsch reference. MAPLE k := 2: with(numtheory): g := 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 0 elif bigomega(n) = 1 and bigomega(pi(n)) = k-1 then sort(expand(x+x*g(pi(n)))) elif bigomega(n) = 1 and bigomega(pi(n)) = k then sort(expand(-x+x*g(pi(n)))) elif bigomega(n) = 1 and bigomega(pi(n)) <> k-1 and bigomega(pi(n)) <> k then sort(expand(x*g(pi(n)))) elif bigomega(s(n)) = k-1 then sort(expand(1+g(r(n))+g(s(n)))) elif bigomega(s(n)) = k then sort(expand(-1+g(r(n))+g(s(n)))) else sort(g(r(n))+g(s(n))) end if end proc; with(numtheory): a := 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 0 elif bigomega(n) = 1 and bigomega(pi(n)) = k-1 then a(pi(n))+subs(x = 1, diff(g(pi(n)), x)) elif bigomega(n) = 1 and bigomega(pi(n)) = k then a(pi(n))-subs(x = 1, diff(g(pi(n)), x)) elif bigomega(n) = 1 and bigomega(pi(n)) <> k and bigomega(pi(n)) <> k-1 then a(pi(n)) elif bigomega(s(n)) = k-1 then a(r(n))+a(s(n))+subs(x = 1, diff(g(r(n))*g(s(n)), x))+subs(x = 1, diff(g(r(n)), x))+subs(x = 1, diff(g(s(n)), x)) elif bigomega(s(n)) = k then a(r(n))+a(s(n))+subs(x = 1, diff(g(r(n))*g(s(n)), x))-subs(x = 1, diff(g(r(n)), x))-subs(x = 1, diff(g(s(n)), x)) else a(r(n))+a(s(n))+subs(x = 1, diff(g(r(n))*g(s(n)), x)) end if end proc: seq(a(n), n = 1 .. 120); CROSSREFS Cf. A206499, A212619, A212620, A212621, A212622, A212623, A212624, A212625, A212626, A212627, A212628, A212629, A212630, A212631, A212632. Sequence in context: A193628 A306506 A241056 * A066602 A073816 A084452 Adjacent sequences:  A212615 A212616 A212617 * A212619 A212620 A212621 KEYWORD nonn AUTHOR Emeric Deutsch, May 22 2012 STATUS approved

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Last modified January 25 03:49 EST 2020. Contains 331241 sequences. (Running on oeis4.)