The teacher writes numbers $1$ at both ends of the blackboard. The first student adds a $2$ in the middle between them, each next student adds the sum of each two adjacent numbers already on the blackboard between them (hence there are numbers $1, 3, 2, 3, 1$ on the blackboard after the second student, $1, 4, 3, 5, 2, 5, 3, 4, 1$ after the third student etc.) Find the sum of all numbers on the blackboard after the $n$-th student.