[seqfan] Re: A120419 "A mysterious sequence"
Paul D Hanna
pauldhanna at juno.com
Tue Aug 26 15:55:07 CEST 2014
Robert,
Your formula simplifies to
E.g.f.: (1 + Series_Reversion( sqrt(2)*( atan(x) + x/(1+x^2) )/2 )^2 )^2.
In PARI code:
A(x) = (1 + serreverse( sqrt(2)*( atan(x) + x/(1+x^2) )/2 )^2 )^2
Can this expression be simplified even more?
Thanks,
Paul
---------- Original Message ----------
From: israel at math.ubc.ca
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Subject: [seqfan] Re: A120419 "A mysterious sequence"
Date: 25 Aug 2014 23:03:26 -0700
On Aug 25 2014, Paul D Hanna wrote:
>Seqfans,
> The "mysterious sequence": https://oeis.org/draft/A120419 has keyword
> "obscure" ...
>
>I found that the e.g.f. satisfies the somewhat obtuse formula:
>
> A(x) = ( 1 + Integral (A(x) * Integral A(x) dx) dx )^2.
>
> The sequence calls for a closed form; can someone derive a closed form
> for this e.g.f.?
>
>Note that the e.g.f. for Euler numbers (A000364) satisfy:
>
> G(x) = 1 + Integral (G(x) * Integral G(x)^2 dx) dx when G(x) =
> 1/cos(x) which makes one suspect that there is a closed form for A120419
> (and perhaps a simpler formula than I gave!).
>
>Regards,
> Paul
>
>_______________________________________________
>
>Seqfan Mailing list - http://list.seqfan.eu/
>
As I wrote in the pink boxes, the E.g.f. A(x) = sum_{i=1}^infty,
a(i)*x^(2*i)/(2*i)! seems to satisfy the implicit equation (for x > 0)
x-(1/2)*sqrt(2)*A(x)*sqrt(sqrt(A(x))-1)*(arctan(sqrt(sqrt(A(x))-1))*sqrt(A(x))+sqrt(sqrt(A(x))-1))/sqrt(A(x)^3*(sqrt(A(x))-1))
= 0
For x < 0 you have to take the other branch of sqrt(sqrt(A(x))-1) I guess,
otherwise the expression in terms of A(x) is an even function.
Not much chance of an "explicit" closed form for A in terms of x.
Cheers,
Robert
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