[seqfan] Re: Fw: Closed form?

[Prof. Dr. Alois Heinz]

Maximilian Hasler schrieb:

>>I have looked at a larger range now, and a(n)/f(n) is drifting slowly
>>upwards or downwards, always with prefix .9233
>>    
>>
>Don't you think that this "drifting up and down" corresponds to the
>periodic variation in the Phi function as per Philippe's comment
>[quoted below], and thus won't disappear ?
>  
>
yes, of course, that is is.  Since Phi is not already encoded into f.

>>Digits:=160:
>>    
>>
>
>^^^ I don't understand this, if we are only interested in the first 10
>digits of the ratio a(n)/f(n).
>Would Maple compute the first 10 digits of f(x) incorrectly without this ?
>  
>
That's right.  At least my (older) version of Maple would not evaluate 
f(x) as a float
if the value of Digits is set too low.  An example:

 > restart;
 > Digits:=100:
 > f:= proc(x) option remember; evalf (sum (x^k/k!/2^binomial(k,2), 
k=0..infinity)) end:
 > f (2^90);

  .225048982437778079564155816798811340218952734375587207935654152\

                                                1101
        3571947332124963929457635315460870670 10

 > restart;
 > Digits:=50:
 > f:= proc(x) option remember; evalf (sum (x^k/k!/2^binomial(k,2), 
k=0..infinity)) end:
 > f (2^90);

                infinity
                 -----                               k
                  \      1237940039285380274899124224
                   )     -----------------------------
                  /               binomial(k, 2)
                 -----        k! 2
                 k = 0


Alois