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 A198336 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. 1
 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, 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, 1, 24, 2, 1, 25, 9, 4, 1, 26, 3, 1, 27, 8, 1, 28, 2, 1, 29, 3, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS This is the pruning operation mentioned, for example, in the Balaban reference (p. 360) and in the Todeschini - Consonni reference (p. 42). 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. First entry of row n is n, last entry is 1. Number of entries in row n is 1 + the radius of the corresponding rooted tree. REFERENCES A. T. Balaban, Chemical graphs, Theoret. Chim. Acta (Berl.) 53, 355-375, 1979. 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 Review, 10, 1968, 273. R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, 2000. LINKS FORMULA 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. EXAMPLE 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. Triangle starts: 1; 2,1; 3,1; 4,1; 5,2,1; 6,2,1; 7,1; 8,1; 9,4,1; 10,3,1; MAPLE 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; CROSSREFS Cf. A198329. Sequence in context: A229994 A165818 A194746 * A330756 A330747 A337785 Adjacent sequences:  A198333 A198334 A198335 * A198337 A198338 A198339 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Dec 01 2011 STATUS approved

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Last modified December 5 14:49 EST 2021. Contains 349557 sequences. (Running on oeis4.)