

A198326


Sum of lengths of all directed paths in the rooted tree having MatulaGoebel 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
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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[j1] for j = 2,3,...,k+1.
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.
a(n)=Sum(k*A198325(n,k), k>=1).


REFERENCES

F. Goebel, On a 11correspondence 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 YeongNan 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..90.
Index entries for sequences related to MatulaGoebel 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 MatulaGoebel 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 MatulaGoebel 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



