From what you wrote, I assume you are considering $n$ and $a_i$ to be positive integers. In this case this equation has a solution if and only if $\gcd(a_1,\ldots,a_k)\mid n$. After removing this common denominator, we may assume that $\gcd(a_1,\ldots,a_k)=1$. In this case, the function $d(n;a_1,\ldots,a_k)$, which counts the number of solutions to this equation, is called the denumerant function of Sylvester. This is related to the famous diophantine problem of Frobenius (see http://en.wikipedia.org/wiki/Coin_problem), which, computationally, is an extremely hard problem.

There is, though, an asymptotics formula for $d(n;a_1,\ldots,a_k)$.

$
d(n;a_1,\ldots,a_k) \approx \dfrac{n^{k-1}}{(k-1)!a_1\ldots a_k}
$, as $n\longrightarrow\infty$

Here is an article on this function http://math.gmu.edu/~geir/SylvDen2.pdf, the first google result of "frobenius problem denumerant asymptotic".