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Consider all arrays of \$\ell\$ non-negative integers in the range \$0,\dots,m\$. The challenge is to work out how many sum to exactly \$s\$.
For \$m=7, \ell = 8\$ the answers for \$s=1 \dots 25\$ the answers are \$1, 8, 36, 120, 330, 792, 1716, 3424, 6371, 11152, 18488, 29184, 44052, 63792, 888...