# [seqfan] Re: Need small amount of help with Apery-like numbers

Hugo Pfoertner yae9911 at gmail.com
Mon Aug 14 12:20:20 CEST 2017

```SeqFans,

I found the discussions between Peter, Joerg and Robert during the
subsequent edits of
https://oeis.org/history?seq=A290576 (2nd sequence cited below) somehow
amusing, e.g. Peter wrote
"*Peter Luschny*: Why on earth has this sequence offset 1 and no one
complains?".

The background of my initial setting a(-1)=0 and a(0)=1 simply was the
definition of the family of sequences provided in the article by A Malik
and A. Straub:

[cite]Similarly the Apery numbers A(n), defined in (1.1), are the solution
of the three term recurrence
(n+1)^3*u(n+1) = (2*n+1)*(a*n^2+a*n+b)*u(n) - n*(c*n^2+d)*u(n-1)
with the choice of parameters a,b,c,d (...) and initial conditions u(-1)=0
and u(0)=1.
[/cite]
Should this general formula be provided in the two new sequences? I had
done the applicable substitutions for a,b,c,d.

I used the complete recurrence formula without change for all results,
starting at n=1, including those for https://oeis.org/A290575 (a=12 b=4
c=16 d=0)
and https://oeis.org/A290576 (a=9 b=3 c=-27 d=0) and provided only the non
trivial terms for n>=1.

If we want the formula above to work "stand-alone" only using the numbers
given in the DATA, then all such sequences
would have to start 0,1,... with offset -1.
Is it common practice to start a sequence with a negative offset to start a
recurrence?

My initial PARI code for A290576 is not defined for n=-1 (all sums empty)
? C=binomial;
? a(n)=sum(k=0, n, sum(l=0, n, C(n, k)^2*C(n, l)*C(k, l)*C(k+l, n)));
? a(0)
%4 = 1
? a(1)
%5 = 3

Hugo Pfoertner

On Mon, Aug 7, 2017 at 2:41 AM, Neil Sloane <njasloane at gmail.com> wrote:

> Hugo,  Thanks so much for doing that.  Very helpful indeed.
>
> I'll take care of the rest - adding the links, etc
> Best regards
> Neil
>
> Neil J. A. Sloane, President, OEIS Foundation.
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Email: njasloane at gmail.com
> >
> > On Sun, Aug 6, 2017 at 10:49 PM, Hugo Pfoertner <yae9911 at gmail.com>
> wrote:
> >
> >> Sequences corresponding to Table 2:
> >>
> >>
> >> # Axxxxxx not in OEIS
> >> # bT2_epsil.txt a=12 b=4 c=16 d=0
> >> 4,40,544,8536,145504,2618176,48943360,941244376,18502137184
> >> <(850)%20213-7184>,370091343040,
> >> 7508629231360,154145664817600,3196100636757760,
> >>
> >> # Axxxxxx not in OEIS
> >> # bT2_zeta_.txt a=9 b=3 c=-27 d=0
> >> 3,27,309,4059,57753,866349,13492251,216077787,3536145057,58875891777,
> >> 994150929951,16984143140589,293036113226223,
>
```