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

[neil greubel]

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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