Smooth Partitons

A smooth partition of the integer \(n\) is a set of positive integers \(a_1, a_2, \ldots a_k\) such that
1. \(k\) is a positive integer,
2. \(a_1 \leq a_2 \leq \cdots \leq a_k,\)
3. \(\sum_{i=1}^{k} a_i = n,\) and
4. \(a_k - a_1 \leq 1.\)
Determine how many smooth partitions there are of the integer \(250.\)

Details and assumptions

Note: The definition given above is not standard terminology.


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