[seqfan] Re: Closed Form Solution to Special Diff Eq?

[Max Alekseyev]

On Fri, Feb 27, 2009 at 5:40 AM, Paul D Hanna <pauldhanna at juno.com> wrote:
> Seqfans,
>      Is there a closed form for the solution A(x) to this differential equation?
>
> 2*A(x)*A(4x) + 8*x*A(x)*A'(4x) - A'(x)*A(4x) = 0
>
> A power series solution exists, but can it be expressed finitely in terms of known functions?

I guess you mean
http://www.research.att.com/~njas/sequences/A155200
that we already discussed in SeqFan a couple of weeks ago.

There is probably no closed-form expressing for the function A(x) but
its coefficients can be expressed in terms of complete Bell
polynomials:
a(n) = B_n( 0!*2^(1^2), 1!*2^(2^2), 2!*2^(2^3), ..., (n-1)!*2^(2^n) ) / n!

btw, could you please point out a proof for the following comment?
%C A155200 More generally, for m integer, exp( Sum_{n>=1} m^(n^2) *
x^n/n ) is a power series in x with integer coefficients.

Thanks,
Max