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A212619 Sum of the distances between all unordered pairs of vertices of degree 3 in the rooted tree with Matula-Goebel number n. 11
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 4, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 3, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,49
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.
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(28)=1 because the rooted tree with Matula-Goebel number 28 is obtained by joining the trees I, I, and Y at their roots; it has 2 vertices of degree 3 (the root and the center of Y), the distance between them is 1.
a(987654321) = 22, as given by the Maple program; the reader can verify this on the rooted tree of Fig. 2 of the Deutsch reference.
MAPLE
k := 3: 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
Sequence in context: A124505 A326855 A025444 * A309162 A345197 A092575
KEYWORD
nonn
AUTHOR
Emeric Deutsch, May 22 2012
STATUS
approved

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Last modified March 19 04:26 EDT 2024. Contains 370952 sequences. (Running on oeis4.)