# [seqfan] Re: Lambert W function and converging factors: merging A001662 and A217538?

Neil Sloane njasloane at gmail.com
Thu Jun 23 08:01:08 CEST 2016

```Paolo,  What you propose sounds excellent.  Please go ahead and make
all those changes!  Thank you.

Note that A001662 has a b-file.  I will go ahead and change the 17th term
to 10125320047141

Fortunately A217538 does not have a b-file, so we don't

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 Wed, Jun 22, 2016 at 5:39 PM, Paolo Bonzini <bonzini at gnu.org> wrote:

>
>
> While preparing a new series for OEIS, I stumbled upon some weirdness in
> the existing series A001662 and A051711:
>
> - A001662 ("Numerators in expansion of W(exp(x)) about x=1, where W is
> the Lambert function") says in the comments section "Please note: a(17)
> is not 10125320047141".  A217538 is the same sequence as A001662 except
> that a(17) is indeed 10125320047141.
>
> - A051711 ("a(0) = 1; for n>0, a(n) = n!*4^n/2") says "For n <= 16,
> denominators in expansion of W(exp(x)) about x=1, where W is the Lambert
> function".
>
>
> From the history, the original definition of A001662 was "Coefficients
> of Airey's converging factor" and, based on Airey's paper, a(17) was
> 10125320047141.  Edit #9 changed it to the current form.  However, I
> believe that the edit was incorrect.
>
> First of all, let's factor the old and new 17-th term:
>
>       10125320047141 = 13*15683*49663379
>         778870772857 =    15683*49663379
>
> This suggests that the coefficient of the series is A001662/A051711, but
> the 17th term of A001662/A051711 is not a reduced fraction.  So as
> things stand A001662 is *not* the coefficient of Airey's converging
> factor, despite saying so in the comments.
>
> coefficients in triangle A008517, evaluated at -1" is wrong.  The values
> in the sequence are simply the polynomials with coefficients in triangle
> A008517, evaluated at -1:
>
>    1-2=                          -1
>    1-8+6=                        -1
>    1-22+58-24=                   13
>    1-52+328-444+120=            -47
>    1-114+1452-4400+3708-720=    -73
>
> without the need for another sign adjustment.  This is also visible from
> the paper "A Sequence of Series for The Lambert W Function" (Section 2.2),
> and is referenced in the comments section of A008517.
>
> Another "yellow flag" is the 1 + 2*x - 9346449274284*x^17 - x*Q(0)
> generating function, where of course the huge x^17 term is simply
> 778870772857-10125320047141.
>
> Luckily, A217638 already has good quality content for the formula,
> maple, mathematica and programs sections.  Therefore, my suggestion is:
>
> 1) change A001662 as follows
>
> - restore the title to "Coefficients of Airey's converging factor"
>
> - restore the 17th term to 10125320047141
>
> - replace the comments section with:
>
>   n!*4^n/2 times the coefficient in expansion of W(exp(x)) about x=1,
>   where W is the Lambert function. - Paolo Bonzini, Jun 22 2016
>
>   The polynomials with coefficients in triangle A008517, evaluated at
>   -1.
>
> - add to the links the following paper (cited in A217538)
>
>   R. M. Corless, D. J. Jeffrey and D. E. Knuth, A sequence of series
>   for the Lambert W Function (section 2.2).
>
>
> http://www.apmaths.uwo.ca/~rcorless/frames/PAPERS/LambertW/CorlessJeffreyKnuth.ps
>
> - replace the formula, maple, mathematica and prog sections of A001662
> with those in A217538
>
>
> 2) delete A217538 which is now a duplicate of A001662 only with fewer
>
> 3) create two new series:
>
> - "Numerators in expansion of W(exp(x)) about x=1, where W is the
> Lambert function, A001662/gcd(A001662,A051711)".  This series can reuse
> most of the content and programs of A001662.
>
> - "Denominators in expansion of W(exp(x)) about x=1, where W is the
> Lambert function, A051711/gcd(A001662,A051711)"
>
>
> If this is okay, I can submit edits for (1) and (3), but I thought it'd
> be clearer to post here first and get approval.
>
> Thanks,
>
> Paolo Bonzini
>
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>
```