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 A058927 Numerators of series related to triangular cacti. 2
 1, 1, 5, 49, 243, 14641, 371293, 253125, 410338673, 16983563041, 1400846643, 41426511213649, 95367431640625, 617673396283947, 10260628712958602189, 756943935220796320321, 7474615974418932603, 827909024473876953125, 456487940826035155404146917, 510798409623548623605717 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS From L. Edson Jeffery, Jan 09 2012: (Start) The reference [Bergeron, et al.] lists the first few terms of the relevant series as S(x) = x + (1/2)*x^3 + (5/8)*x^5 + (49/48)*x^7 + (243/128)*x^9 + ..., from which the numerators were taken for this sequence and the denominators for A058928. This leads to the following Conjecture: S(x) = Sum_{n>=0} ((2*n+1)^(n-1)/(n!*2^n))*x^(2*n+1) = (A052750(n)/A000165(n))*x^(2*n+1). Letting D_n be the set of divisors of n! and d_n = max(k in D_n : k | (2*n+1)^(n-1)), then a(n)=A052750(n)/d_n. (End) The above conjecture is correct and follows from formula given in A034940 for the number of rooted labeled triangular cacti with 2n+1 nodes. - Andrew Howroyd, Aug 30 2018 REFERENCES F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 307. LINKS Andrew Howroyd, Table of n, a(n) for n = 0..200 FORMULA G.f.: A(x) satisfies A(x)=exp(x*A(x)^2). - Vladimir Kruchinin, Feb 09 2013 a(n) = numerator(A034940(n)/(2*n+1)!) = numerator((2*n+1)^(n-1)/(2^n*n!)). - Andrew Howroyd, Aug 30 2018 PROG (PARI) a(n)={numerator((2*n+1)^(n-1)/(2^n*n!))} \\ Andrew Howroyd, Aug 30 2018 CROSSREFS Cf. A000165, A034940, A052750, A058928. Sequence in context: A007406 A196326 A273385 * A083224 A201358 A242035 Adjacent sequences:  A058924 A058925 A058926 * A058928 A058929 A058930 KEYWORD nonn,frac,easy AUTHOR N. J. A. Sloane, Jan 12 2001 EXTENSIONS More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 25 2010 Terms a(12) and beyond from Andrew Howroyd, Aug 30 2018 STATUS approved

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Last modified March 31 22:16 EDT 2020. Contains 333151 sequences. (Running on oeis4.)