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%I #13 Apr 09 2013 09:37:43
%S 1,0,1,2,0,0,1,1,1,0,2,3,0,2,1,0,1,0,0,0,1,2,1,0,1,1,1,2,0,1,1,4,0,0,
%T 2,1,2,0,3,2,0,1,0,3,1,0,0,1,0,0,2,3,1,0,0,2,1,1,1,0,3,2,2,0,1,0,1,1,
%U 1,1,0,0,0,0,1,5,0,1,0,1,1,0,2,0,2,1,2,2,0,2,1,1,3,0,2,1,3,0,1,0,0,1,1,1,3,0,1,2,2,0,0,1,0,2,1
%N Triangle read by rows: T(n,k) is the number of leaves at level k>=1 of the rooted tree having Matula-Goebel number n (n>=2).
%C 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.
%C Number of entries in row n is A109082(n) (n=2,3,...).
%D F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
%D I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
%D I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
%D D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%F We give the recursive construction of the row generating polynomials P(n)=P(n,x). P(2)=x; if n = p(t) (=the t-th prime), then P(n)=x*P(t); if n=rs (r,s>=2), then P(n)=P(r)+P(s) (2nd Maple program yields P(n)).
%e Row n=7 is [0,2] because the rooted tree with Matula-Goebel number 7 is the rooted tree Y, having 0 leaves at level 1 and 2 leaves at level 2.
%e Row n=2^m is [m] because the rooted tree with Matula-Goebel number 2^m is a star with m edges.
%e Triangle starts:
%e 1;
%e 0,1;
%e 2;
%e 0,0,1;
%e 1,1;
%e 0,2;
%e 3;
%e 0,2;
%p with(numtheory): P := 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 = 2 then x elif bigomega(n) = 1 then sort(expand(x*P(pi(n)))) else sort(P(r(n))+P(s(n))) end if end proc: for n from 2 to 30 do seq(coeff(P(n), x, k), k = 1 .. degree(P(n))) end do; # yields sequence in triangular form
%p with(numtheory): P := 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 = 2 then x elif bigomega(n) = 1 then sort(expand(x*P(pi(n)))) else sort(P(r(n))+P(s(n))) end if end proc: for n from 2 to 30 do P(n) end do;
%Y Cf. A109082.
%K nonn,tabf
%O 2,4
%A _Emeric Deutsch_, Oct 06 2011
%E Keyword tabf added by _Michel Marcus_, Apr 09 2013
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