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 A193405 The Matula numbers of the rooted trees that have a perfect matching. 3
 2, 5, 6, 15, 18, 22, 23, 26, 31, 41, 45, 54, 55, 65, 66, 69, 78, 93, 94, 103, 122, 123, 135, 137, 158, 162, 165, 166, 167, 195, 198, 202, 207, 211, 234, 235, 242, 253, 254, 279, 282, 283, 286, 299, 305, 309, 338, 341, 358, 366, 369, 394, 395, 401, 403, 405, 411, 415, 419, 431, 451, 474, 486, 495, 498 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. It is known that a tree has at most one perfect matching. Complement of A193406. 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. C. D. Godsil, Algebraic Combinatorics, Chapman & Hall, New York, 1993. LINKS E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288. FORMULA Define b(n) (c(n)) to be the generating polynomials of the matchings of the rooted tree with Matula-Goebel number n that contain (do not contain) the root, with respect to the size of the matching. We have the following recurrence for the pair M(n)=[b(n),c(n)]. M(1)=[0,1]; if n=p(t) (=the t-th prime), then M(n)=[xc(t),b(t)+c(t)]; if n=rs (r,s,>=2), then M(n)=[b(r)c(s)+c(r)b(s), c(r)c(s)]. Then m(n)=b(n)+c(n) is the generating polynomial of the matchings of the rooted tree with respect to the size of the matchings (a modified matching polynomial). The tree has a perfect matching if and only if the degree of this polynomial is 1/2 of the number of vertices of the tree. EXAMPLE 2,6,31 are in the sequence because they are the Matula numbers of the paths on 2,4,6 vertices, respectively. MAPLE with(numtheory): N := 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 1 elif bigomega(n) = 1 then 1+N(pi(n)) else N(r(n))+N(s(n))-1 end if end proc: M := 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 [x*M(pi(n))[2], M(pi(n))[1]+M(pi(n))[2]] else [M(r(n))[1]*M(s(n))[2]+M(r(n))[2]*M(s(n))[1], M(r(n))[2]*M(s(n))[2]] end if end proc: m := proc (n) options operator, arrow: sort(expand(M(n)[1]+M(n)[2])) end proc: PM := {}: for n to 500 do if N(n) = 2*degree(m(n)) then PM := `union`(PM, {n}) else  end if end do: PM; CROSSREFS Cf. A061775, A193406. Sequence in context: A287203 A291211 A193402 * A037079 A101325 A042980 Adjacent sequences:  A193402 A193403 A193404 * A193406 A193407 A193408 KEYWORD nonn AUTHOR Emeric Deutsch, Feb 12 2012 STATUS approved

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Last modified January 16 23:44 EST 2019. Contains 319206 sequences. (Running on oeis4.)