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A198338 Irregular triangle read by rows: row n is the sequence of Matula numbers of the rooted subtrees of the rooted tree with Matula-Goebel 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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.

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 n-1 and it ends with the first occurrence of n.

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..85.

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

FORMULA

Row 1 is [1]; if n = p(t) (= the t-th 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 Matula-Goebel 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

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Last modified May 7 17:40 EDT 2021. Contains 343652 sequences. (Running on oeis4.)