Consider distinct arrangements of $k$ nonnegative integers that sum to $s$ with the the additional condition that the sum of every other integer is $t \le s$. A bijective mapping of these arrangements to multisets is given by the prefix sum of the first $k-1$ integers. What is the total number of permutations over all multisets thus obtained?
For example, when $k=4, s=3,$ and $t=2$, the mapping of arrangements to multisets is \begin{pmatrix} (2,1,0,0) & (2,0,0,1) & (1,1,1,0) & (1,0,1,1) & (0,1,2,0) & (0,0,2,1)\\ \{2,3,3\} & \{2,2,2\} & \{1,2,3\} & \{1,1,2\} & \{0,1,3\} & \{0,0,2\} \end{pmatrix} and the number of permutations over all multisets is $3+1+6+3+6+3=22$.