[seqfan] Re: The Demichel errors, continued

D. S. McNeil dsm054 at gmail.com
Sat Dec 17 22:03:21 CET 2011

"We are a museum. In the basement are some fossils with the wrong labels,
fossils hitherto unknown to science. We should throw them away
and replace them with the correct fossils?!"



Anyway, I scanned through all the sequences in a013538.txt and found
64 which I couldn't roughly identify from the g.f. given in the name.
(By roughly, I mean ignore a few at the start -- for example, A012509
seems to be missing an initial 2, etc. -- and don't care about the
offsets or signs.)

Of the "unknown", 41 involve csc/csch.  0 of the 1477 known sequences
involved csc.

Following up on NJAS' earlier suggestion, I chose a few test sequences
and experimented with scanning all expressions (w/o essential
singularities at zero) of the form f(g(x) [+-] h(x)) and f(g(h(x))),
with f,g,h in

       [sin, cos, tan, sinh, cosh, tanh,
         sec, sech, csc, csch, cot, coth,
         asin, asinh, acos, acosh, atan, atanh,
         asec, asech, acsc, acsch, acot, acoth,
         exp, log, sqrt]

but couldn't find anything which worked, either because there's
nothing to be found or because of an oversight on my part.

Here are the 64 sequences in a013538.txt which weren't trivially
recognized by my code.  I don't claim they're wrong, much less
inexplicably so, only that they passed my first filter and that I'm
pretty sure I can identify the remaining ones, so those interested in
playing guess-the-sequence can concentrate their efforts below:

A012854 log(log(x + 1)*csc(x))
A012856 log(arcsin(x)*csc(x))
A012857 log(arctan(x)*csc(x))
A012858 log(sinh(x)*csc(x))
A012859 log(arcsinh(x)*csc(x))
A012860 log(tanh(x)*csc(x))
A012861 log(arctanh(x)*csc(x))
A012862 arccosh(arctanh(x)*csc(x))
A012863 log(log(x + 1)*cot(x))
A012865 log(arcsin(x)*cot(x))
A012867 log(arctan(x)*cot(x))
A012868 log(arcsinh(x)*cot(x))
A012869 arccos(arcsinh(x)*cot(x))
A012870 log(tanh(x)*cot(x))
A012871 log(arctanh(x)*cot(x))
A013305 arccosh(e^x - sin(x))
A013306 arccosh(e^x - arcsin(x))
A013307 arccosh(e^x - tan(x))
A013308 arccosh(e^x - arctan(x))
A013323 arccosh(e^x - arcsinh(x))
A013324 arccosh(e^x - tanh(x))
A013325 arccosh(e^x - arctanh(x))
A013516 e^(csc(x) - cot(x))
A013517 sin(csc(x) - cot(x))
A013518 arcsin(csc(x) - cot(x))
A013519 tan(csc(x) - cot(x))
A013521 cos(csc(x) - cot(x))
A013522 sinh(csc(x) - cot(x))
A013523 arcsinh(csc(x) - cot(x))
A013524 tanh(csc(x) - cot(x))
A013526 cosh(csc(x) - cot(x))
A013527 sec(csc(x) - cot(x))
A013528 sech(csc(x) - cot(x))
A013529 e^(csc(x) - csch(x))
A013530 sin(csc(x) - csch(x))
A013531 arcsin(csc(x) - csch(x))
A013532 tan(csc(x) - csch(x))
A013533 arctan(csc(x) - csch(x))
A013534 cos(csc(x) - csch(x))
A013535 sec(csc(x) - csch(x))
A013536 e^(csc(x) - coth(x))
A013537 sin(csc(x) - coth(x))
A013538 arcsin(csc(x) - coth(x))
A013539 tan(csc(x) - coth(x))
A013540 arctan(csc(x) - coth(x))
A013541 cos(csc(x) - coth(x))
A013542 sinh(csc(x) - coth(x))
A013543 arcsinh(csc(x) - coth(x))
A013544 tanh(csc(x) - coth(x))
A013545 arctanh(csc(x) - coth(x))
A013546 cosh(csc(x) - coth(x))
A013547 sec(csc(x) - coth(x))
A013548 sech(csc(x) - coth(x))
A013550 e^(cot(x) - csch(x))
A013551 e^(cot(x) - coth(x))
A013552 sin(cot(x) - coth(x))
A013553 -arcsin(cot(x) - coth(x))
A013554 -tan(cot(x) - coth(x))
A013555 arctan(cot(x) - coth(x))
A013556 cos(cot(x) - coth(x))
A013557 sec(cot(x) - coth(x))
A013559 e^(-coth(x) + csch(x))
A013568 log(arcsin(x)/log(x + 1))
A013570 log(arctan(x)/log(x + 1))


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