# [seqfan] The Demichel errors, continued

N. J. A. Sloane njas at research.att.com
Sat Dec 17 20:39:01 CET 2011

```Dear Seq Fans,
Richard Mathar and I have been looking into the problem.
One clue is that sequences involving cosec(x) are often (alays?)
wrong.

Suppose back in 1996 there was a bug in the csc function.
I looked for the first cosec in his list (see a013538,txt), and it is here:

%I A012854
%S A012854 1,0,0,55,7,38,219,1487,11570,101682,995254,1814866833,126624718,
%T A012854 1619876114,22344623929,95542242048111,5222279193717,87722551872998,
%U A012854 1561364880901471,29353667049630175,581233980566953138
%V A012854 -1,0,-0,55,-7,38,-219,1487,-11570,101682,-995254,1814866833,
%W A012854 -126624718,1619876114,-22344623929,95542242048111,-5222279193717,
%X A012854 87722551872998,-1561364880901471,29353667049630175
%N A012854 log(cosec(x)*log(x+1))=-x+55/4!*x^4-7/5!*x^5+38/6!*x^6-219/7!*x^7...
%R A012854
%O A012854 0,4
%K A012854 sign,done
%A A012854 Patrick DEMICHEL (dml at hpfrcu03.france.hp.com)

Sure enough, this is wrong!

If we assume the log parts of log(cosec(x)*log(x+1)) are correct,
then this says that - if my calculations are correct -  that

series(1*x^(-1)-1/2-1/12*x+5/24*x^2+1541/720*x^3-1601/1440*x^4-10187/60480*x^5+18337/
40320*x^6+1186291/518400*x^7-1793257/1451520*x^8-11568989/68428800*x^9+475543283/
958003200*x^10+14112561699877/2615348736000*x^11-375261235627/134120448000*x^12-\
106979176397/251073478656*x^13+14384882695861/12553673932800*x^14+39233031299960999/
2910169866240000*x^15-63754358387747431/9146248151040000*x^16-54613060385569779241/
51090942171709440000*x^17+19469639890894136927/6812125622894592000*x^18+O(x^19),x,19)

or in other words (I don't know is this will work in email)

-1                        2   1541  3   1601  4   10187  5   18337  6   1186291  7
s5 := x   - 1/2 - 1/12 x + 5/24 x  + ---- x  - ---- x  - ----- x  + ----- x  + ------- x
720       1440      60480      40320      518400

1793257  8   11568989  9   475543283  10   14112561699877  11   375261235627  12
- ------- x  - -------- x  + --------- x   + -------------- x   - ------------ x
1451520      68428800      958003200       2615348736000        134120448000

106979176397  13   14384882695861  14   39233031299960999  15   63754358387747431
- ------------ x   + -------------- x   + ----------------- x   - -----------------
251073478656       12553673932800       2910169866240000        9146248151040000

16   54613060385569779241  17   19469639890894136927  18      19
x   - -------------------- x   + -------------------- x   + O(x  )
51090942171709440000       6812125622894592000

Unfortunately this does not explain the other errors.
For example,
%I A012856
%S A012856 6,40,1197,888,6945664,976514838,836054516,47528446559105,
%T A012856 15241280332373681,13709908797232583,2906353877902946339671,
%U A012856 2663198791987064856806,1535992958057301680879410
%N A012856 log(cosec(x)*arcsin(x))=6/2!*x^2+40/4!*x^4+1197/6!*x^6+888/8!*x^8...
%R A012856
%O A012856 0,1
%K A012856 nonn
%A A012856 Patrick DEMICHEL (dml at hpfrcu03.france.hp.com)

is still wrong, even if we replace cosec by bad-cosec.

Yet this one is correct:
%I A012063
%S A012063 1,1,1,3,9,49,249,1963,14129,146017,1362961,17450227,200409465,
%T A012063 3064381137,41812246345,744091967643,11765081428705,239039270010305,
%U A012063 4298599985014945,98237724213189475,1980239469431883625
%N A012063 exp(arcsin(arcsin(x)))=1+x+1/2!*x^2+3/3!*x^3+9/4!*x^4+49/5!*x^5...
%R A012063
%O A012063 0,4
%K A012063 nonn
%A A012063 Patrick DEMICHEL (dml at hpfrcu03.france.hp.com)

so presumably arcsin is OK.

So we don't understand what is going on yet.

Maybe some seq fan can solve the puzzle.

Neil

```