A058927 Numerators of series related to triangular cacti.

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

Offset: 0

Author

N. J. A. Sloane, Jan 12 2001

Keywords

• nonn
• frac
• easy

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

Crossrefs

Cf. A000165, A034940, A052750, A058928.

Programs

PARI

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

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

Extensions

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

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