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A058928 Denominators of series related to triangular cacti.

%I A058928 #17 Jun 04 2020 11:45:43
%S A058928 1,2,8,48,128,3840,46080,14336,10321920,185794560,6553600,81749606400,
%T A058928 78479622144,209924915200,1428329123020800,42849873690624000,
%U A058928 170993385472000,7611536747003904,1678343852714360832000,747740921331712000,2551082656125828464640000
%N A058928 Denominators of series related to triangular cacti.
%C A058928 From _L. Edson Jeffery_, Jan 09 2012: (Start)
%C A058928 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 denominators were taken for this sequence and the numerators for A058927. This leads to the following
%C A058928 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)=A000165(n)/d_n. (End)
%C A058928 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
%D A058928 F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 307.
%H A058928 Andrew Howroyd, <a href="/A058928/b058928.txt">Table of n, a(n) for n = 0..200</a>
%F A058928 a(n) = denominator(A034940(n)/(2*n+1)!) = denominator((2*n+1)^(n-1)/(2^n*n!)). - _Andrew Howroyd_, Aug 30 2018
%o A058928 (PARI) a(n)={denominator((2*n+1)^(n-1)/(2^n*n!))} \\ _Andrew Howroyd_, Aug 30 2018
%Y A058928 Cf. A000165, A034940, A052750, A058927.
%K A058928 nonn,frac,easy
%O A058928 0,2
%A A058928 _N. J. A. Sloane_, Jan 12 2001
%E A058928 More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 25 2010
%E A058928 Terms a(12) and beyond from _Andrew Howroyd_, Aug 30 2018