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

[Paolo Bonzini]

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