[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
the bad-cosec function is
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
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