sequence arising from asymptotic expansion

[benoit]

Let z(k) be defined as follows :

z(0)=4^n ;  z(k)=4^(n-k)+sqrt(z(k-1))

Let  S(n)=sum(k=0,n,1/z(k))

Let  T(n)=sum(k=0,n,1/(2^k+1)^2)

I found experimentaly :

S(n)-T(n) = 2/8^n + 13/16^n + 132/32^n + 4283/64^n + 260102/128^n + 
32852729 /256^n+ 8341600776 /512^n+O(1/1024^n)

How the sequence of numerators continues ? Is there any pattern?


Benoit Cloitre

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