[seqfan] More about the errors in Patrick Demichel's sequences
N. J. A. Sloane
njas at research.att.com
Sat Dec 17 19:34:11 CET 2011
I have been trying to guess what went wrong, without success.
Sequences A013500-A013515 are correct. The first error in this set is
at A013516, where the definition is said to be
exp(cosec(x)-cotan(x))=1+x+1/5!*x^5+2/6!*x^6+7/7!*x^7+23/8!*x^8...
But this eqaution is not true. Call the right-hand side p(x).
We know 26 terms of p(x).
One obvious guess is that cosec(x)-cotan(x) is correct and "exp" is wrong.
But cosec(x)-cotan(x) = 1/2*x+1/24*x^3+1/240*x^5+17/40320*x^7+31/725760*x^9+691/159667200*x^11+...
, so what map sends that to p(x)???
The other guess is that exp is correct and cosec(x)-cotan(x) is wrong.
So let's look at log p(x). This is
x-1/2*x^2+1/3*x^3-1/4*x^4+5/24*x^5-31/180*x^6+151/1008*x^7-5297/40320*x^8+...
which is an e.g.f. with coefficients
0, 1, -1, 2, -6, 25, -124, 755, -5297, 42714, ... What is this sequence???
Neil
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