I do not have a concise formula for $c(n)$, but it can be more or less easily computed from the expansion of
$$\prod_{1\leq i<j\leq n} (a_j - a_{j-1} - (a_i - a_{i-1})),$$
where we conveniently define $a_0 := 0$. Namely, we are concerned only about terms of multi-degree being some permutation of $\{0,1,\dots,n-1\}$, and $c(n)$ equals the sum of the coefficients of such terms multiplied by their multi-degree (permutation) signs.

For example, when $n=3$ the product yields the following terms of interest:
$$-3 a^{(2, 1, 0)} - 3 a^{(1, 2, 0)} -3 a^{(0, 2, 1)} - 2a^{(1, 0, 2)} + a^{(0, 1, 2)},$$
which give $$c(3) = (-3)(-1) + (-3)(+1) + (-3)(-1) + (-2)(-1) + 1(+1) = 6.$$

Here is a [sample Sage code](https://sagecell.sagemath.org/?z=eJxNT7FuhTAM3PkKb88pAZUOHSrxD--xIoYITGQEDkqCRP--gQyvHiyd73xnTzTDiKJ-CkjVQQtPt_6K29isHYvF10uLfpwPdQvOJOg_Byhh5RCxqy1JQJXJPZG7d9MxRgQ8-2WoUquaQUF1Yb4w33h2HhZgAW_EEn5pKZs85fe00YuCbO0pHl4gHBuC-XiS345oIjvBnsvmvRjriawnSjcNqg5sBbOt0fHid-AZgvORJvynVW17_5NzRaVYKIoRv9UfBa9VUA==&lang=sage&interacts=eJyLjgUAARUAuQ==) for computing $c(n)$.