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%I #8 Oct 12 2012 15:00:58
%S 1,2,1,3,1,4,1,5,2,1,6,2,1,7,1,8,1,9,4,1,10,3,1,11,3,1,12,2,1,13,2,1,
%T 14,2,1,15,6,2,1,16,1,17,2,1,18,4,1,19,1,20,3,1,21,4,1,22,5,2,1,23,4,
%U 1,24,2,1,25,9,4,1,26,3,1,27,8,1,28,2,1,29,3,1
%N Irregular triangle read by rows: row n is the sequence of Matula-Goebel numbers of the rooted trees obtained from the rooted tree with Matula-Goebel number n by pruning it successively 0,1,2,... times. The operation of pruning consists of the removal of the vertices of degree one, together with their incident vertices.
%C This is the pruning operation mentioned, for example, in the Balaban reference (p. 360) and in the Todeschini - Consonni reference (p. 42).
%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 First entry of row n is n, last entry is 1.
%C Number of entries in row n is 1 + the radius of the corresponding rooted tree.
%D A. T. Balaban, Chemical graphs, Theoret. Chim. Acta (Berl.) 53, 355-375, 1979.
%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.
%D R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, 2000.
%F A198329(n) is the Matula-Goebel number of the rooted tree obtained by removing from the rooted tree with Matula-Goebel number n the vertices of degree one, together with their incident edges. Repeated application of this yields the Matula-Goebel numbers of the trees obtained by successive prunings. The Maple program is based on this.
%e Row 7 is 7, 1 because the rooted tree with Matula-Goebel number 7 is Y and after the first pruning we obtain the 1-vertex tree having Matula-Goebel number 1. Row 5 is 5, 2, 1 because it refers to the path tree on four vertices; after pruning it becomes the 1-edge tree with Matula-Goebel number 2.
%e Triangle starts:
%e 1;
%e 2,1;
%e 3,1;
%e 4,1;
%e 5,2,1;
%e 6,2,1;
%e 7,1;
%e 8,1;
%e 9,4,1;
%e 10,3,1;
%p with(numtheory): a := proc (n) local r, s, b: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: b := proc (n) if n = 1 then 1 elif n = 2 then 1 elif bigomega(n) = 1 then ithprime(b(pi(n))) else b(r(n))*b(s(n)) end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then b(pi(n)) else b(r(n))*b(s(n)) end if end proc: S := proc (m) local A, i: A[m, 1] := m: for i while a(A[m, i]) < A[m, i] do A[m, i+1] := a(A[m, i]) end do: seq(A[m, j], j = 1 .. i) end proc: for n to 15 do S(n) end do;
%Y Cf. A198329.
%K nonn,tabf
%O 1,2
%A _Emeric Deutsch_, Dec 01 2011
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