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 A190175 The Goebel-Matula numbers of the rooted trees having only vertices of odd degree. 1
 2, 7, 8, 28, 32, 43, 53, 98, 112, 128, 172, 212, 227, 263, 311, 343, 392, 443, 448, 512, 577, 602, 688, 742, 848, 908, 1052, 1193, 1244, 1372, 1423, 1568, 1619, 1772, 1792, 1993, 2048, 2107, 2308, 2311, 2408, 2539, 2597, 2752, 2939, 2968, 3178, 3209, 3392, 3632, 3682, 3698, 3779 (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. 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. LINKS E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 FORMULA The number of vertices of even degree of the rooted trees with Matula-Goebel number n is A190174(n). The number n is in the sequence if and only if A190174(n)=0. In A182907 one can find the generating polynomial g(n)=g(n,x) of the vertices of the rooted tree having Matula-Goebel number n, according to degree. We look for those values of n for which the polynomial g(n,x) is odd, i.e. satisfies g(n,-x)=-g(n,x). EXAMPLE 7 is in the sequence because the rooted tree with Matula-Goebel number 7 is the rooted tree Y with vertices of degree 1,1,1,3. 15 is not in the sequence because the rooted tree with Matula-Goebel number 15 is the path tree ABRCDE, rooted at R; it has 2 vertices of degree 1 and 4 vertices of degree 2. 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 1 elif bigomega(n) = 1 then sort(expand(g(pi(n))+x^bigomega(pi(n))*(x-1)+x)) else sort(expand(g(r(n))+g(s(n))-x^bigomega(r(n))-x^bigomega(s(n))+x^bigomega(n))) end if end proc: a := proc (n) options operator, arrow: (1/2)*subs(x = 1, g(n))+(1/2)*subs(x = -1, g(n)) end proc: A := {}: for n to 4000 do if a(n) = 0 then A := `union`(A, {n}) else  end if end do: A; CROSSREFS Cf. A182907, A190174. Sequence in context: A117558 A117559 A236291 * A081700 A093795 A001493 Adjacent sequences:  A190172 A190173 A190174 * A190176 A190177 A190178 KEYWORD nonn AUTHOR Emeric Deutsch, Oct 30 2011, Dec 09 2011 STATUS approved

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Last modified April 13 16:24 EDT 2021. Contains 342936 sequences. (Running on oeis4.)