15. Let $b_{i}=a_{i}-i+1 \quad(i=1,2, \cdots, k)$, then $1 \leqslant b_{1} \leqslant b_{2} \leqslant \cdots \leqslant b_{k} \leqslant n-k$ +1, and each $b_{i}(i=1,2, \cdots, k)$ is an odd number; conversely, this is also true, so the number of solutions is equal to the $k$-combinations with repetition of all odd numbers in $1,2, \cdots, n-k+1$ (a total of $m=\left[\frac{n-k+2}{2}\right]$ numbers), hence it is $\mathrm{C}_{m+k-1}^{k}$.
15. Let $k$ be a fixed positive integer, $1 \leqslant k \leqslant n$. Determine the number of integer sequences $1 \leqslant a_{1}$ $<a_{2}<\cdots<a_{k} \leqslant n$ that satisfy $a_{i}$ and $i$ have the same parity $(i=1,2, \cdots, k)$.