Putnam

Prompt

Let n be a positive integer. For i,j in {1,2,...,n}, let s(i,j) be the number of pairs (a,b) of nonnegative integers satisfying ai + bj = n. Let S be the n-by-n matrix whose (i,j)-entry is s(i,j). Determine det(S) as an explicit function of n. Provide a complete, rigorous proof valid for every positive integer n. Numerical experimentation alone is insufficient. Do not consult external sources.