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

0,3

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 A352371 A201358

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