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%I #15 Mar 07 2017 11:31:54
%S 1,1,1,1,1,1,2,1,1,1,1,1,2,2,2,1,1,3,1,2,2,1,2,2,1,1,1,1,1,1,3,2,2,2,
%T 1,3,2,2,2,2,4,1,2,1,1,1,3,3,3,1,1,3,2,1,3,2,1,2,2,1,1,2,2,1,1,4,2,2,
%U 2,2,1,3,3,3,3,1,4,2,2,2,2,3,3,1,1,1,1,1,1,1,5,1,2,2,2,1,3,2,1,3,2,2,4,3,3,2,1,4,2,3,3,1,4,2,1
%N Irregular triangle read by rows: T(n,k) is the number of vertices having escape distance k>=0 in the rooted tree having Matula-Goebel number n.
%C The escape distance of a vertex v in a rooted tree T is the distance from v to the nearest leaf of T that is a descendant of v. For the rooted tree ARBCDEF, rooted at R, the escape distance of B is 4 (the leaf A is not a descendant of B).
%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 Each row is nonincreasing (each vertex with escape distance k (k>=1) is the parent of some vertex with escape distance k-1).
%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.
%H E. Deutsch, <a href="http://arxiv.org/abs/1111.4288">Tree statistics from Matula numbers</a>, arXiv preprint arXiv:1111.4288, 2011
%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F We give the recursive construction of the row generating polynomials P(n)=P(n,x): if n = p(t) (=the t-th prime), then P(n)=P(t)+x^{1+LLL(t)}; if n=rs (r,s>=2), then P(n)=P(r)+P(s)-x^{max(LLL(r),LLL(s))}; LLL denotes the level of the lowest leaf (computed recursively and programmed in A184166) (2nd Maple program yields P(n)).
%e Row n=7 is [2,1,1] because the rooted tree with Matula-Goebel number 7 is the rooted tree Y, having 2 leaves and 1 (1) vertex at distance 1 (2) from either of the leaves.
%e Triangle starts:
%e 1;
%e 1,1;
%e 1,1,1;
%e 2,1;
%e 1,1,1,1;
%e 2,2;
%e 2,1,1;
%p with(numtheory): P := proc (n) local r, s, LLL: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: LLL := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+LLL(pi(n)) else min(LLL(r(n)), LLL(s(n))) end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then P(pi(n))+x^(1+LLL(pi(n))) else P(r(n))+P(s(n))-x^max(LLL(r(n)), LLL(s(n))) end if end proc: for n to 30 do seq(coeff(P(n), x, k), k = 0 .. degree(P(n))) end do; # yields sequence in triangular form
%p with(numtheory): P := proc (n) local r, s, LLL: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: LLL := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+LLL(pi(n)) else min(LLL(r(n)), LLL(s(n))) end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then P(pi(n))+x^(1+LLL(pi(n))) else P(r(n))+P(s(n))-x^max(LLL(r(n)), LLL(s(n))) end if end proc: P(998877665544);
%Y Cf. A184170.
%K nonn,tabf
%O 1,7
%A _Emeric Deutsch_, Oct 23 2011
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