second sequence arising from asymptotic expansion

[benoit]

z(k) is still defined as follows :

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 ...? 
Thanks.


Benoit Cloitre
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