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A198326 Sum of lengths of all directed paths in the rooted tree having Matula-Goebel number n. 1
0, 1, 4, 2, 10, 5, 7, 3, 8, 11, 20, 6, 13, 8, 14, 4, 16, 9, 10, 12, 11, 21, 19, 7, 20, 14, 12, 9, 23, 15, 35, 5, 24, 17, 17, 10, 16, 11, 17, 13, 26, 12, 19, 22, 18, 20, 29, 8, 14, 21, 20, 15, 13, 13, 30, 10, 14, 24, 30, 16, 22, 36, 15, 6, 23, 25, 22, 18, 23, 18, 26, 11, 25, 17, 24, 12, 27, 18, 38, 14, 16, 27, 36, 13, 26, 20, 27, 23, 19, 19 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

A directed path of length k in a rooted tree is a sequence of k+1 vertices v[1], v[2], ..., v[k], v[k+1], such that v[j] is a child of v[j-1] for j = 2,3,...,k+1.

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.

a(n)=Sum(k*A198325(n,k), k>=1).

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.

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..90.

Index entries for sequences related to Matula-Goebel numbers

FORMULA

In A198325 we give the recursive construction of the generating polynomials P(n)=P(n,x) of the directed paths of the rooted tree corresponding to the Matula-Goebel number n, with respect to length. a(n) is the derivative dP(n,x)/dx, evaluated at x=1.

EXAMPLE

a(7)=7 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y, having 3 directed paths of length 1 (the edges) and 2 directed paths of length 2.

MAPLE

with(numtheory): P := proc (n) local r, s, E: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: E := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+E(pi(n)) else E(r(n))+E(s(n)) end if end proc: if n = 1 then 0 elif bigomega(n) = 1 then sort(expand(x*E(n)+x*P(pi(n)))) else sort(P(r(n))+P(s(n))) end if end proc: seq(subs(x = 1, diff(P(n), x)), n = 1 .. 90);

CROSSREFS

Cf. A198325

Sequence in context: A247414 A138569 A191725 * A283938 A283943 A283942

Adjacent sequences:  A198323 A198324 A198325 * A198327 A198328 A198329

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Nov 02 2011

STATUS

approved

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Last modified January 20 02:05 EST 2019. Contains 319320 sequences. (Running on oeis4.)