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

Sean A. Irvine sairvin at gmail.com
Thu Aug 9 03:14:52 CEST 2018


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.
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>



More information about the SeqFan mailing list