# [seqfan] Re: The Ramanujan alpha, beta, and gamma series

Ron Knott ron at ronknott.com
Tue May 17 15:41:11 CEST 2016

```The “a” series A051028 has GF
(1 + 53 x + 9 x^2)/(1 - 82 x - 82 x^2 + x^3) = 1 + 135 x + 11161 x^2 + 926271 x^3 + 76869289 x^4+…
and replacing x by 1/x gives the “alpha” series
(x (9 + 53 x + x^2))/(1 - 82 x - 82 x^2 + x^3).
The x factor is irrelevant as a GF.  Note that we have just reversed the coefficients in the numerator and this is the GF of
(9 + 53 x + x^2)/(1 - 82 x - 82 x^2 + x^3) = 9 + 791 x + 65601 x^2 + 5444135 x^3 + 451797561 x^4+… i.e. A272853
So perhaps the GFs should be changed to this form to be consistent with the rest of OEIS and the second form used in A272853?

Similarly for the betas: A051030
(2 + 8 x - 10 x^2)/(1 - 82 x - 82 x^2 + x^3) = 2 + 172 x + 14258 x^2 + 1183258 x^3 + 98196140 x^4+…
and
(-10 + 8 x + 2 x^2)/(1 - 82 x - 82 x^2 + x^3) = -10 - 812 x - 67402 x^2 - 5593538 x^3 - 464196268 x^4
Negating this  series to make all coefficients positive gives A272854

Similarly
(2 - 26 x - 12 x^2)/(1 - 82 x - 82 x^2 + x^3) = 2 + 138 x + 11468 x^2 + 951690 x^3 + 78978818 x^4 +…
which is A051029
and again reversing the denominator’s coefficients (ignoring the x factor) gives
(-12 - 26 x + 2 x^2)/(1 - 82 x - 82 x^2 + x^3) = -12 - 1010 x - 83802 x^2 - 6954572 x^3 - 577145658 x^4
the series A272855

Perhaps these changes might clear up some confusion between these 6 series and make things more consistent with the rest of OEIS?

Ron Knott

> On 9 May 2016, at 02:51, Robert Munafo <mrob27 at gmail.com> wrote:
>
> Neil F., thanks for checking that my Wolfram Alpha thing also works in MMA.
> Always good to know.
>
> Ron K., A050787 - A050794 are relevant (though different), thanks. I added
> mine upon observing that the number 3111529740489 (for example) didn't show
> up in bing/google/etc. (3111529740489^3+3196919629018^3=3974802064140^3+1)
> I still think someone must have done that Laurent series expansion
> somewhere.
>
> Giovanni R., thank you. I did try adding "CoefficientList[..., x]" around
> it like we see in A051028, but Wolfam Alpha wouldn't accept that. I'd blame
> that human-friendly fuzzy-matching parsing, like you said.
>
> --
>  Robert Munafo  --  mrob.com
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/

```

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