second sequence arising from asymptotic expansion

[Paul D Hanna]

    The sequence of numerators that you derived, 
if taken as coefficients of a power series -A(x), 
has a fourth-root that remains an integer series
(at least for the first 9 terms): 

A(x) = {1,-4,22,-144,1173,-12540,183930,-3760128,107083642,...}

A(x)^(1/2) = {1,-2,9,-54,438,-4908,76749,-1658742,49172642,...}

A(x)^(1/4) = {1,-1,4,-23,188,-2174,35184,-781167,23596744,...}

A superficial observation, but it would be curious (and unlikely) 
should this continued to hold for higher-order terms. 
    Paul

On Fri, 27 Jun 2003 21:04:34 +0200 benoit <abcloitre at wanadoo.fr> writes:
...
> z(0)=4^n ;  z(k)=4^(n-k)+sqrt(z(k-1))
> 
> Let  U(n)=sum(k=0,n,sqrt(z(k)))
> 
> it appears that :
> 
> U(n)-2^(n+1)-n+1+1/2^n =
> 
>   -1/8^n + 4/16^n - 22/32^n + 144/64^n -1173/128^n + 
> 12540/256^n-183930/512^n +3760128/1024^n+O(1/2048^n)
> 
> And this second sequence of numerators is :
> 
> -1, 4 , -22, 144, -1173, 12540, -183930, 3760128, -107083642 ....
> 
> Can anyone supply more terms for this one too, comment, formula ...? 
...
> Benoit Cloitre