Sequence Defined In Terms Of EVERY Term In It

[Leroy Quet]

I posted this to sci.math. I am asking about any sequence in general. But 
for the purpose of seq.fan, I am asking about integer sequences.

(I am uncertain if the {a[]} sequence I refer to below is an integer 
sequence or not. But if it IS an integer sequence this would pose some 
problems with convergence of the product and  some of the sums.)

--------------------------------

I am wondering about any good examples of infinite sequences where
every terms depends on every other term (prior AND following), and
where there is only one such sequence which exists given the rule that
generated it.

(I would suppose that some of these sequences may be proved to exist,
but are incalcuable.)

An example of such a sequence (calcuable/incalcuable?), IF this
sequence is unique and it exists:

{a[k]} is such that:

sum{k=0 to oo} a[k] *x^k =

product{k=0 to oo} (1 +a[k]*x),

where the product and sum converge for all x in some positive-lengthed
interval.

(Maybe there is a way to, say, recursively generate this sequence in
principle, but then the sum and/or product might not converge for any
x but x=0.)

Some facts:

* a[0] = 1. This is the only term I know of as of now.

* a[1] = sum{j=0 to oo} a[j].

* a[k] = sum of all products of k a[]'s, where the indexes in each
product are distinct.

* a[1] also = sum{j=0 to oo} a[j]/(1 +a[j])

* and so 0 = sum{j=0 to oo} a[j]^2/(1 +a[j]).

* So, if all a[]'s are real and the most recent sum converges,
 then some a[]'s are < -1.

Thanks,
Leroy Quet