EDIT: The vectors can be any length. That is $k$ is not fixed.

For a given natural number $n$, let $S\_1(n)$ be the number of vectors $(a\_1, a\_2, \ldots, a\_k)$ such that

$$a\_1 + a\_2 + \cdots + a\_k = n$$

where each $a\_i$ is an odd natural number. What is the value of $S\_1(n)$? Is there a closed form solution?

### A Variation

Suppose we relax the condition on vectors so that all $a\_i$ must be odd except $a\_1$ and $a\_k$, which can be either even or odd. Call the number of such vectors $S\_2(n)$. What is the value of $S\_2(n)$? Is there a closed form solution?

### Notes

This question was inspired by a recent [post](https://math.stackexchange.com/questions/1373679/how-many-different-sums-of-parts-of-a-vector/1373691#1373691) by [phoenix](https://math.stackexchange.com/users/91386/phoenix), where we are asked a question about vectors. I have figured out some bounds for $SO$ in that post. In particular, it is between $\sqrt{2^{n - 1}}$ and $2^{n - 1}$. Note that $2^{n - 1}$ is the number of vectors adding to $n$ without the odd restriction.