[seqfan] Re: Cute interpretation of A002294

jean-paul allouche jean-paul.allouche at imj-prg.fr
Fri Aug 11 20:49:21 CEST 2023

Dear Allan

Though I have not looked carefully enough, it seems
that this is what the sentence
 From Stillwell (1995), p. 62: "Eisenstein's Theorem. If y^5 + y = x, 
then y has a power series expansion y = x - x^5 + 10*x^9/2^1 - 15 * 14 * 
x^13/3! + 20 * 19 * 18*x^17/4! - ...."
probably means (when I said I did not look carefully,
I mean that I am not sure that I have understood
where the minus signs comme from, and that I
have only checked the (absolute value of the)
first coefficients in the power series above.

best wishes

Le 09/08/2023 à 03:29, Allan Wechsler a écrit :
> According to the Wikipedia article about "Bring radicals", (plus a tiny bit
> of elementary algebra),
> the generating function for A002294 is the inverse function of f(x) = x^5 +
> x. I can't see this mentioned anywhere in the A002294 entry, but perhaps
> I'm being blind, and I would appreciate another pair of eyes
> double-checking me before I add it.
> If you adjoin this generating function to the usual toolkit of addition,
> subtraction, multiplication, division, and exponentiation, then, pace the
> Abel-Ruffini theorem, you *can* construct a quintic formula, with which any
> quintic can be solved. (Of course the formula is disastrously complicated.
> I have not seen it written out.)
> --
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