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 A212623 Irregular triangle read by rows: T(n,k) is the number of independent vertex subsets with k vertices of the rooted tree with Matula-Goebel number n (n>=1, k>=0). 10
 1, 1, 1, 2, 1, 3, 1, 1, 3, 1, 1, 4, 3, 1, 4, 3, 1, 4, 3, 1, 1, 4, 3, 1, 1, 5, 6, 1, 1, 5, 6, 1, 1, 5, 6, 1, 1, 5, 6, 2, 1, 5, 6, 2, 1, 5, 6, 2, 1, 6, 10, 4, 1, 5, 6, 4, 1, 1, 5, 6, 2, 1, 6, 10, 5, 1, 5, 6, 4, 1, 1, 6, 10, 5, 1, 1, 6, 10, 5, 1, 1, 6, 10, 4, 1, 6, 10, 5, 1, 6, 10, 7, 2, 1, 7, 15 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS A vertex subset in a tree is said to be independent if no pair of vertices is connected by an edge. The empty set is considered to be independent. 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. Sum of entries in row n = A184165(n) = number of independent vertex subset (the Merrifield-Simmons index). Sum(k*T(n,k), k>=0) = A212624(n) = number of vertices in all independent vertex subsets. Number of entries in row n = 1 + number of vertices in the largest independent vertex susbset = 1 + A212625(n). Last entry in row n = A212626(n) = number of largest independent vertex subsets. With the given Maple program, the command P(n) yields the generating polynomial of row n. REFERENCES H. Prodinger and R. F. Tichy, Fibonacci numbers of graphs, Fibonacci Quart., 20, 1982, 16-21. 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 Y-N. 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. LINKS E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288. FORMULA Define R(n) =R(n,x)  (S(n)=S(n,x)) the generating polynomial of the  independent vertex subsets that contain (do not contain) the root of the rooted tree with Matula-Goebel number n. Then R(1)=x, S(1)=1, R(the t-th prime) = x*S(t), S(the t-th prime) = R(t) + S(t); R(rs) = R(r)R(s)/x, S(rs) = S(r)S(s), (r,s>=2). EXAMPLE Row 5 is [1,4,3] because the rooted tree with Matula-Goebel number 5 is the path tree R - A - B - C with independent vertex subsets: {}, {R}, {A}, {B}, {C}, {R,B}, {R,C}, {A,C}. Triangle starts: 1, 1; 1, 2; 1, 3, 1; 1, 3, 1; 1, 4, 3; MAPLE with(numtheory): A := 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 = 1 then [x, 1] elif bigomega(n) = 1 then [expand(x*A(pi(n))[2]), expand(A(pi(n))[1])+A(pi(n))[2]] else [sort(expand(A(r(n))[1]*A(s(n))[1]/x)), sort(expand(A(r(n))[2]*A(s(n))[2]))] end if end proc: P := proc (n) options operator, arrow: sort(A(n)[1]+A(n)[2]) end proc: for n to 35 do seq(coeff(P(n), x, k), k = 0 .. degree(P(n))) end do; % yields sequence in triangular form CROSSREFS Cf. A212618, A212619, A212620, A212621, A212622, A212624, A212625, A212626, A212627, A212628, A212629, A212630, A212631, A212632. Sequence in context: A065371 A300982 A186007 * A229214 A218578 A006346 Adjacent sequences:  A212620 A212621 A212622 * A212624 A212625 A212626 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Jun 01 2012 STATUS approved

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Last modified April 11 06:11 EDT 2021. Contains 342886 sequences. (Running on oeis4.)