[seqfan] Re: An old sequence that seems to be wrong

[Neil Sloane]

Sean,  Thank you for the analysis of A013568.  Your argument
is convincing.  I have edited that entry, and I created A317209 for the
corrected values.
It is good to have this old mystery cleared up!

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 Thu, Aug 9, 2018 at 2:19 AM, neil greubel <jthomae at gmail.com> wrote:

> Does the Maple program also produce selected values of A278562 and
> A278563 ?   In my previous attempts to determine the correct matching
> of the expansion I tried to change the ArcSin to ArcSinh, ArcTanh,
> ArcTan, ArcCos, ArcCosh. Also attempted to change Log for Exp in
> various combinations and powers. From A27856(2/3) it is given that
> log(arcsin(x)/log(1+x)) = (1/2)*x - (1/24)*x^2 + (1/8)*x^3 -
> (5/192)*x^4 + (19/288)*x^5 - (6863/362880)*x^6 + ... .
> This is what makes A013568 more difficult to determine.
>
> On Wed, Aug 8, 2018 at 9:14 PM Sean A. Irvine <sairvin at gmail.com> wrote:
> >
> > In 2016 it was noted that the series expansion in A013568 did not match
> the
> > description.  Today I tried to resolve this mystery and made some
> > progress.  I believe the sequence really is a failed attempt to give the
> > e.g.f. expansion of log(asin(x)/log(1+x).
> >
> > First the offset in A013568 is probably wrong and likely should be 1
> (this
> > is not that surprising since I think Demichel added them all with offset
> 0
> > originally).
> >
> > Define in Maple:
> >
> > f:=n->trunc(n!*coeff(series(log(arcsin(x)/log(x+1)),x,n+1),x^n));
> >
> > and compare the values with the existing entries.  It's not a perfect
> match
> > to the listed values (misses for n=1, 6, 10, 18), but they are
> sufficiently
> > similar that I think Demichel really was trying to expand
> > log(arcsin(x)/log(x+1) in some manner, but perhaps mistakenly thought the
> > coefficients would be integers after multiplication by n!.
> >
> > A013568(n)         f(n)
> > 1                  0
> > 0                  0
> > 0                  0
> > 0                  0
> > 7                  7
> > -665               -13
> > 219                219
> > -598               -598
> > 11570              11570
> > -5357968           -44280
> > 995254             995254
> > -4960662           -4960662
> > 126624718          126624718
> > -783821598         -783821598
> > 22344623929        22344623929
> > -166137700653      -166137700653
> > 5222279193717      5222279193717
> > -16426560893778654 -45502938763929
> > 1561364880901471   1561364880901471
> > -15643758252397591 -15643758252397591
> >
> > Sean.
> >
> >
> > On 24 November 2016 at 06:02, Neil Sloane <njasloane at gmail.com> wrote:
> >
> > > Dear SeqFans, Many years ago Patrick Demichel
> > > sent in a large number of Taylor series expansions that he obtained
> > > by taking all combinations of log / exp / sin / arcsin / ...
> > > of depth <= 3.
> > > G. C. Greubel and Joerg Arndt recently noticed that A013568 does not
> match
> > > its description.
> > > which is
> > > log(arcsin(x)/log(x+1))=x+7/5!*x^5-665/6!*x^6+219/7!*x^7-
> 598/8!*x^8...,
> > > so this is a mystery. Can anyone figure out what this is the Taylor
> > > expansion of?
> > > 1, 0, 0, 0, 7, -665, 219, -598, 11570, -5357968, 995254, -4960662,
> > > 126624718, -783821598, 22344623929, -166137700653, 5222279193717,
> > > -16426560893778654, 1561364880901471, -15643758252397591
> > >
> > > One possibility would be to look at all the other depth-three
> > > Taylor series that he submitted, and see which one is missing!
> > > But perhaps an inspired guess would be quicker? I tried taking the exp
> > > of his expression, but that did not seem to help.
> > >
> > > --
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> > >
> >
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