

A182907


Triangle read by rows: row n is the degree sequence (written in nondecreasing order) of the rooted tree with MatulaGoebel number n.


6



0, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 3, 1, 1, 1, 2, 3, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 4, 1, 1, 1, 2, 3, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 4, 1, 1, 1, 2, 2, 3, 1, 1, 1, 2, 2, 3, 1, 1, 2, 2, 2, 2, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 2, 4, 1, 1, 2, 2, 2, 2, 2
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OFFSET

1,6


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.
The number of entries in row n is A061775(n) (= number of vertices of the rooted tree).


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..119.
E. Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288, 2011


FORMULA

For a graph with degree sequence a,b,c,..., define the degree sequence polynomial to be x^a + x^b + x^c + ... . The degree sequence polynomial g(n)=g(n,x) of the rooted tree with MatulaGoebel number n can be obtained recursively in the following way: g(1)=1; if n=p(t) (=the tth prime), then g(n)=g(t)+x^G(t)*(x1)+x; if n=rs (r,s>=2), then g(n)=g(r)+g(s)x^G(r)x^G(s)+x^G(n); G(m) is the number of prime divisors of m counted with multiplicities. The Maple program, based on this recursive procedure, finds for an arbitrary n the polynomial g(n,x) and then extracts from this polynomial the degree sequence S(n).


EXAMPLE

Row 7 is 1,1,1,3 because the rooted tree with MatulaGoebel number 7 is the rooted tree Y.
Row 32 is 1,1,1,1,1,5 because the rooted tree with MatulaGoebel number 32 is a star with 5 edges.


MAPLE

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 1 elif bigomega(n) = 1 then sort(expand(g(pi(n))+x^bigomega(pi(n))*(x1)+x)) else sort(expand(g(r(n))+g(s(n))x^bigomega(r(n))x^bigomega(s(n))+x^bigomega(n))) end if end proc: S := proc (n) if n = 1 then 0 else seq(seq(j, i = 1 .. coeff(g(n), x, j)), j = 1 .. degree(g(n))) end if end proc: for n to 25 do S(n) end do; # yields sequence in triangular form


CROSSREFS

A061775
Sequence in context: A050412 A307017 A220424 * A334745 A323231 A175128
Adjacent sequences: A182904 A182905 A182906 * A182908 A182909 A182910


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Oct 05 2011


STATUS

approved



