[seqfan] Re: A058927

[Ed Jeffery]

Seqfans,

Replying to my own post:

>From the reference,

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

in Neil Sloane's sequences A058927 <https://oeis.org/A058927> and
A058928<https://oeis.org/A058928>,
is shown the series

(1) Delta = x+(1/2)*x^3+(5/8)*x^5+(49/48)*x^7+(243/128)*x^9+....

The numerators of the coefficients in (1) are given by

A058927={1,1,5,49,243,...}

and the denominators by

A058928={1,2,8,48,128,...}.

However, I believe that (1) is just the series

(2) Delta = Sum[n=0,...,infinity, ((2*n+1)^(n-1)/(n!*2^n))*x^(2*n+1)]

in disguise, in which the numerators of the coefficients are given by

A052750 <https://oeis.org/A052750> = {1,1,5,49,729,14641,371293,11390625,
...}

and the denominators by

A000165 <https://oeis.org/A000165> = {1,2,8,48,384,3840,46080,645120,...}.

The discrepancy arises from the fact that some of the coefficients in (2)
factor (so the fractions can be reduced), e.g., 729/384=243/128, accounting
for the differences of entries between A058927 and A052750 as well as
between A058928 and A000165. Maybe someone else would like to verify this
hypothesis to make sure it is correct.

Regards,

Ed Jeffery