Double summation limits

Can anyone please explain me why it is not coming? I have done  
following :

j=1
i=0:j; sum( choose(i+j,i) )

[1] 3

j=2
i=0:j; sum( choose(i+j,i) )

[1] 10

j=3
i=0:j; sum( choose(i+j,i) )

[1] 35

j=0:3; i=0:j; sum( choose(i+j,i) )

Warning message:
In 0:j : numerical expression has 4 elements: only the first used
[1] 4

Here if value of "j" is given independently it is ok. However as  
soon as "j"
is used in loop it fails?

Why it is happening? Is there any mathematics behind that?

Orestis Chrysafis wrote:

Dear R experts
I need to write a function that incorporates double summation, the  
problem
being that the upper limit of the second summation is the index of  
the
first
summation, i.e:

sum_{j=0}^{x} sum_{i=0}^{j} choose(i+j, i)

where x variable or constant, doesn't matter.
The following code obviously doesn't work:

f=function(x) {j=0:x; i=0:j; sum( choose(i+j,i) )  }

Can you help?
Thanks in advance,

Orestis Chrysafis
School of Mathematics
University of Sheffield