[seqfan] Re: What constant is involved in creating this cf?

[Olivier Gerard]

On Wed, Jul 24, 2013 at 9:48 AM, <allouche at math.jussieu.fr> wrote:

> Hi
>
> Expansions of powers of e and "combinations" of those are more
> or less classical. See, e.g.,
> http://en.wikipedia.org/wiki/Continued_fraction
> and more precisely
>
> http://en.wikipedia.org/wiki/Continued_fraction#Regular_patterns_in_continued_fractions
>
> We find there in particular
> tanh(1/n) = [0 ; n, 3n , 5n, 7n, ...]
> hence tanh(1) = [0 ; 1, 3, 5, 7, ...]
>

Dan, as Jean-Paul wrote, these ones are known since Euler, at least.

I was wondering what your question was exactly.

By adjusting with rationals and inversion,
one can have the sequence starts where one wants.

For instance

1/tanh(1) = (e^2+1)/(e^2-1) gives [1; 3, 5, 7, 9, 11, ...]
(e^2-1)/2  gives [3; 5, 7, 9, 11, ...]
2/(e^2-7)  gives your [5; 7, 9, 11, 13, ...]
(e^2-7)/(37-5e^2) gives [7; 9, 11, 13, 15, ...]
(37-5e^2)/(36e^2-266) gives [7; 9, 11, 13, 15, ...]

and you can go on as far as you want.

Coefficients of the rational fractions are given by

A001515 Bessel polynomial y_n(1)

1, 2, 7, 37, 266, 2431, 27007, 353522, 5329837, 90960751, 1733584106,

for the constant coefficient  (the link to CFs is mentionned by Benoit
Cloitre
in this sequence).

and
A000806 Bessel polynomial y_n(-1)

1, 0, 1, -5, 36, -329, 3655, -47844, 721315, -12310199, 234615096

for the coefficient of e^2

and this sequence count among other things sets of pairs
of integers without successive integers.


Olivier