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 for computing $c(n)$.

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. (Notice that only such terms do not vanish under alternating summation over $\sigma$. Furthermore, the summation of each such term gives $V$ multiplied by the sign of the term multi-degree.)

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 for computing $c(n)$.

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: $$-3 a^{(3, 2, 1)} - 3 a^{(2, 3, 1)} -3 a^{(1, 3, 2)} - 2a^{(2, 1, 3)} + a^{(1, 2, 3)},$$ which give $$c(3) = (-3)(-1) + (-3)(+1) + (-3)(-1) + (-2)(-1) + 1(+1) = 6.$$

Here is a sample Sage code for computing $c(n)$.

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 for computing $c(n)$.