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%I #17 May 07 2015 09:22:16
%S 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,
%T 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,
%U 1,2,2,3,1,1,2,2,4,1,4,2,3,1,1
%N 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.
%C 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.
%C Number of entries in row n is the number of prime divisors of n counted with multiplicity.
%C Sum of entries in row n = A196052(n).
%H E. Deutsch, <a href="http://arxiv.org/abs/1111.4288">Rooted tree statistics from Matula numbers</a>, arXiv:1111.4288 [math.CO], 2011.
%H E. Deutsch, <a href="http://dx.doi.org/10.1016/j.dam.2012.05.012">Rooted tree statistics from Matula numbers</a>, Discrete Appl. Math., 160, 2012, 2314-2322.
%H F. Goebel, <a href="http://dx.doi.org/10.1016/0095-8956(80)90049-0">On a 1-1-correspondence between rooted trees and natural numbers</a>, J. Combin. Theory, B 29 (1980), 141-143.
%H I. Gutman and A. Ivic, <a href="http://dx.doi.org/10.1016/0012-365X(95)00182-V">On Matula numbers</a>, Discrete Math., 150, 1996, 131-142.
%H I. Gutman and Yeong-Nan Yeh, <a href="http://www.emis.de/journals/PIMB/067/3.html">Deducing properties of trees from their Matula numbers</a>, Publ. Inst. Math., 53 (67), 1993, 17-22.
%H D. W. Matula, <a href="http://www.jstor.org/stable/2027327">A natural rooted tree enumeration by prime factorization</a>, SIAM Rev. 10 (1968) 273.
%F 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.
%e 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.
%e Triangle starts:
%e 0;
%e 1;
%e 2;
%e 1,1;
%e 2;
%e 1,2;
%e 3;
%e 1,1,1;
%p 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);
%Y Cf. A196052, A001222.
%K nonn,tabf
%O 1,3
%A _Emeric Deutsch_, May 04 2015