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 A257537 Number of subtrees with at least one edge of the rooted tree with Matula-Goebel number n. 0
 0, 1, 3, 3, 6, 6, 7, 7, 10, 10, 10, 12, 12, 12, 15, 15, 12, 19, 15, 18, 18, 15, 19, 24, 21, 19, 29, 22, 18, 27, 15, 31, 21, 18, 25, 37, 24, 24, 27, 34, 19, 33, 22, 25, 40, 29, 27, 48, 30, 37, 25, 33, 31, 56, 28, 42, 34, 27, 18, 51, 37, 21, 49, 63, 36, 36, 24, 30, 40, 45, 34, 73, 33, 37, 54, 42, 33, 48, 25 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS The Matula 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  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 numbers of the m branches of T. a(n) = A184161(n) - A061775(n). The subtree polynomial of a tree T is the bivariate generating polynomial of the subtrees of T with at least one edge with respect to the number of edges (marked by x) and number of pendant vertices (marked by y). See the Martin et. al, reference. For example, the subtree polynomial of the tree \/ is 2xy^2 + x^2 y^2. If G(n;x,y) is the subtree polynomial of the rooted tree with Matula number n, then, obviously, a(n) = G(n;1,1). The Maple program uses this circuitous way for finding a(n). With the given Maple program, the command G(n) yields the subtree polynomial of the rooted tree having Matula number n (this is my "secret" reason to include this sequence). LINKS E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011. E. Deutsch, Rooted tree statistics from Matula numbers, Discrete Appl. Math., 160, 2012, 2314-2322. 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 Rev. 10 (1968) 273. EXAMPLE a((4)=3 because the rooted tree with Matula number 4 is  \/ with subtrees \ ,  / , and  \/ . MAPLE with(numtheory): g := 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 elif bigomega(n) = 1 then expand(x*y^2+x*g(pi(n))+x*y*h(pi(n))) else expand(g(r(n))+g(s(n))) end if end proc: h := 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 elif bigomega(n) = 1 then 0 else expand(h(r(n))+h(s(n))+g(r(n))*g(s(n))/y^2+g(r(n))*h(s(n))/y+h(r(n))*g(s(n))/y+h(r(n))*h(s(n))) end if end proc: G := 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 elif bigomega(n) = 1 then g(n)+G(pi(n)) else expand(G(r(n))+G(s(n))+h(n)-h(r(n))-h(s(n))) end if end proc: seq(subs({x = 1, y = 1}, G(i)), i = 1 .. 150); CROSSREFS Cf. A184161, A061775 Sequence in context: A327142 A175394 A070318 * A219883 A219381 A219852 Adjacent sequences:  A257534 A257535 A257536 * A257538 A257539 A257540 KEYWORD nonn AUTHOR Emeric Deutsch, Apr 28 2015 STATUS approved

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Last modified October 14 02:29 EDT 2019. Contains 327995 sequences. (Running on oeis4.)