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 A193404 Number of matchings (independent edge subsets) in the rooted tree with Matula-Goebel number n. 0
 1, 2, 3, 3, 5, 5, 4, 4, 8, 8, 8, 7, 7, 7, 13, 5, 7, 12, 5, 11, 11, 13, 12, 9, 21, 12, 20, 10, 11, 19, 13, 6, 21, 11, 18, 16, 9, 9, 19, 14, 12, 17, 10, 18, 32, 20, 19, 11, 15, 30, 18, 17, 6, 28, 34, 13, 14, 19, 11, 25, 16, 21, 28, 7, 31, 31, 9, 15, 32, 27, 14 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS A matching in a graph is a set of edges, no two of which have a vertex in common. The empty set is considered to be a matching. 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. REFERENCES 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. É. Czabarka, L. Székely, and S. Wagner, The inverse problem for certain tree parameters, Discrete Appl. Math., 157, 2009, 3314-3319. LINKS E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288. FORMULA Define b(n) (c(n)) to be the number of matchings of the rooted tree with Matula-Goebel number n that contain (do not contain) the root. We have the following recurrence for the pair A(n)=[b(n),c(n)]. A(1)=[0,1]; if n=p(t) (=the t-th prime), then A(n)=[c(t),b(t)+c(t)]; if n=rs (r,s,>=2), then A(n)=[b(r)c(s)+c(r)b(s), c(r)c(s)]. Clearly, a(n)=b(n)+c(n). See the Czabarka et al. reference (p. 3315, (2)). The Maple program is based on this recursive formula. EXAMPLE a(3)=3 because the rooted tree with Matula-Goebel number 3 is the path ABC on 3 vertices; it has 3 matchings: empty, {AB}, {BC}. 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 [0, 1] elif bigomega(n) = 1 then [A(pi(n))[2], A(pi(n))[1]+A(pi(n))[2]] else [A(r(n))[1]*A(s(n))[2]+A(s(n))[1]*A(r(n))[2], A(r(n))[2]*A(s(n))[2]] end if end proc: a := proc (n) options operator, arrow: A(n)[1]+A(n)[2] end proc: seq(a(n), n = 1 .. 80); CROSSREFS Cf. A184165. Sequence in context: A046146 A081768 A273493 * A072923 A257003 A131922 Adjacent sequences:  A193401 A193402 A193403 * A193405 A193406 A193407 KEYWORD nonn AUTHOR Emeric Deutsch, Feb 11 2012 STATUS approved

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Last modified January 18 23:05 EST 2019. Contains 319282 sequences. (Running on oeis4.)