This isn't in the homework section so I'm assuming it isn't homework.
Let A be the set of sequences of the type in the post.
Let B be the set of sequences of N+1 non-negative integers with sum N-2.
A and B are in bijection by associating a sequence [itex]a_1,a_2, \ldots ,a_N[/itex] of A to [itex]b_1,b_2,\ldots,b_{N+1}[/itex] defined by [itex]b_i = a_i - a_{i-1}[/itex] where we let [itex]a_0 = 1, a_{N+1}=N[/itex]. The inverse is given by associating a sequence [itex]b_1,b_2,\ldots,b_{N+1}[/itex] of B to [itex]a_1,\ldots,a_N[/itex] defined by [itex]a_{i+1} = a_{i}+b_{i+1}[/itex] where we let [itex]a_0 = 1[/itex].
Thus we can just count B which has
[tex]\binom{(N+1)+(N-2)-1}{(N+1)-1} = \binom{2N-2}{N}[/tex]
elements.
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