

A198338


Irregular triangle read by rows: row n is the sequence of Matula numbers of the rooted subtrees of the rooted tree with MatulaGoebel number n. A root subtree of a rooted tree T is a subtree of T containing the root.


1



1, 1, 2, 1, 2, 3, 1, 2, 2, 4, 1, 2, 3, 5, 1, 2, 2, 3, 4, 6, 1, 2, 3, 3, 7, 1, 2, 2, 2, 4, 4, 4, 8, 1, 2, 2, 3, 3, 4, 6, 6, 9, 1, 2, 2, 3, 4, 5, 6, 10, 1, 2, 3, 5, 11, 1, 2, 2, 2, 3, 4, 4, 4, 6, 6, 8, 12, 1, 2, 3, 3, 4, 6, 6, 7, 15, 1, 2, 2, 3, 3, 4, 5, 6, 6
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OFFSET

1,3


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.
Number of entries in row n is A184160(n). Row n>=2 can be easily identified: it starts with the entry following the first occurrence of n1 and it ends with the first occurrence of n.


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..85.
E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288


FORMULA

Row 1 is [1]; if n = p(t) (= the tth prime), then row n is [1, p(a), p(b), ... ], where [a,b,...] is row t; if n=rs (r,s >=2), then row n consists of the numbers r[i]*s[j], where [r[1], r[2],...] is row r and [s[1], s[2], ...] is row s. The Maple program, based on this recursive procedure, yields row n (<=1000; adjustable) with the command MRST(n).


EXAMPLE

Row 4 is [1,2,2,4] because the rooted tree with MatulaGoebel number 4 is V and its root subtrees are *, , , and V.
Triangle starts:
1;
1,2;
1,2,3;
1,2,2,4;
1,2,3,5;
1,2,2,3,4,6;


MAPLE

with(numtheory): MRST := 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([1, seq(ithprime(mrst[pi(n)][i]), i = 1 .. nops(mrst[pi(n)]))]) else sort([seq(seq(mrst[r(n)][i]*mrst[s(n)][j], i = 1 .. nops(mrst[r(n)])), j = 1 .. nops(mrst[s(n)]))]) end if end proc: for n to 1000 do mrst[n] := MRST(n) end do;


CROSSREFS

Cf. A198339.
Sequence in context: A238966 A334230 A275723 * A199086 A098053 A272907
Adjacent sequences: A198335 A198336 A198337 * A198339 A198340 A198341


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Dec 04 2011


STATUS

approved



