[seqfan] Re: A120419 "A mysterious sequence"

[israel at math.ubc.ca]

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
>
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>

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