A257540 Irregular triangle read by rows: row n (n>=2) contains the degrees of the level 1 vertices of the rooted tree having Matula-Goebel number n; row 1: 0.

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

Offset: 1

Author

Emeric Deutsch, May 04 2015

Keywords

• nonn
• tabf

Comments

The Matula (or 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 the number of prime divisors of n counted with multiplicity.

Sum of entries in row n = A196052(n).

Examples

			Row 8 is 1,1,1. Indeed, the rooted tree with Matula number 8 is the star tree \|/; vertices at level 1 have degrees 1,1,1.
Triangle starts:
0;
1;
2;
1,1;
2;
1,2;
3;
1,1,1;
		

Links

• E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
• E. Deutsch, Rooted tree statistics from Matula numbers, Discrete Appl. Math., 160, 2012, 2314-2322.
• 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 Rev. 10 (1968) 273.

Crossrefs

Cf. A196052, A001222.

Programs

Maple

with(numtheory): DL := proc (n) if n = 2 then [1] elif bigomega(n) = 1 then [1+bigomega(pi(n))] else [op(DL(op(1, factorset(n)))), op(DL(n/op(1, factorset(n))))] end if end proc: with(numtheory): DL := proc (n) if n = 1 then [0] elif n = 2 then [1] elif bigomega(n) = 1 then [1+bigomega(pi(n))] else [op(DL(op(1, factorset(n)))), op(DL(n/op(1, factorset(n))))] end if end proc: seq(op(DL(n)), n = 1 .. 100);

Formula

Denoting by DL(n) the multiset of the degrees of the level 1 vertices of the rooted tree with Matula number n, we have DL(1)=[0], DL[2]=[1], DL(i-th prime) = [1+bigomega(i)], DL(rs) = DL(r) union DL(s), where bigomega(i) is the number of prime divisors of i, counted with multiplicity (A001222) and "union" is "multiset union". The Maple program is based on these recurrence equations.