A058928 -id:A058928 - cp's OEIS frontend

1, 1, 5, 49, 243, 14641, 371293, 253125, 410338673, 16983563041, 1400846643, 41426511213649, 95367431640625, 617673396283947, 10260628712958602189, 756943935220796320321, 7474615974418932603, 827909024473876953125, 456487940826035155404146917, 510798409623548623605717

Offset: 0

N. J. A. Sloane, Jan 12 2001

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

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 307.

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A000165, A034940, A052750, A058928.

a(n)={numerator((2*n+1)^(n-1)/(2^n*n!))} \\ Andrew Howroyd, Aug 30 2018

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

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 25 2010

Terms a(12) and beyond from Andrew Howroyd, Aug 30 2018

cp's OEIS Frontend

A058927 Numerators of series related to triangular cacti.

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