

A212618


Sum of the distances between all unordered pairs of vertices of degree 2 in the rooted tree with MatulaGoebel 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
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OFFSET

1,9


COMMENTS

The MatulaGoebel number of a rooted tree can be 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.


REFERENCES

F. Goebel, On a 11 correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
I. Gutman and YN. Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.


LINKS

Table of n, a(n) for n=1..94.
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.
Index entries for sequences related to MatulaGoebel numbers


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 MatulaGoebel number n with respect to level. We have a(1) = 0; if n = p(t) (=the tth prime) and bigomega(t)=k1 then a(n) = a(t)+[dg(t)/dx]_{x=1}; if n = p(t) (=the tth prime) and bigomega(t)=k, then a(n) = a(t)[dg(t)/dx]_{x=1}; if n = p(t) (=the tth prime) and bigomega(t) =/ k and =/ k1, then a(n)= a(t); if n = rs with r prime, s>=2, bigomega(s)=k1, 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) =/ k1 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 MatulaGoebel 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)) = k1 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)) <> k1 and bigomega(pi(n)) <> k then sort(expand(x*g(pi(n)))) elif bigomega(s(n)) = k1 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)) = k1 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)) <> k1 then a(pi(n)) elif bigomega(s(n)) = k1 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: A233581 A193628 A241056 * A066602 A073816 A084452
Adjacent sequences: A212615 A212616 A212617 * A212619 A212620 A212621


KEYWORD

nonn


AUTHOR

Emeric Deutsch, May 22 2012


STATUS

approved



