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

[Neil Sloane]

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
have to worry about that.



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.
Phone: 732 828 6098; home page: http://NeilSloane.com
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.
>
> In addition, in the comments, "(-1)^n times the polynomials with
> 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
> links and references.
>
> 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
>
>
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