How to determine the no. of integral partitions into $k$ parts?

Question

I wanted to know, if I was to partition $500$ into positive $k$ integers, not necessarily distinct under the following constraints

1.k is +ve.

2.all k parts need not be distinct.

3.the first integer subtracted from the last integer in the k parts is smaller than or equal to 1.( a(k)-a(1)<=1)

what would be a formula for the number of possibilities?

Any help appreciated.

Thanks.

Answer 1

You simply have this expression

$$a_1+a_2+ \dots a_k=500$$

Now it's just a Stars and Bars problem. Assuming $a_i$'s are all non-negative integers as Alex says. You have infinitely many ways to partition if you let negative integers.