TITLE: Determinantal questions on Alternate Sign Matrices
QUESTION [6 upvotes]: Let $\mathcal{A}_n$ be the set of all Alternating Sign Matrices (ASM) of size $n\times n$. The cardinality $\#\mathcal{A}_n$ is well-known
$$\#\mathcal{A}_n=\prod_{k=0}^{n-1} \frac{(3k+1)!}{(n+k)!}.$$
I brought up some questions to bear.

QUESTION 1: What are these values on the sum of determinants?
  $$\sum_{M\in\mathcal{A}_n}\det(M).$$
QUESTION 2: What is known (can be said) about the determinants distribution of these matrices in $\mathcal{A}_n$? Such as, (a) what is the range of possible values? (b) how many are singular/regular? (c) how many achieve the value 1? or -1?

EXAMPLES. Define the Laurent polynomial
$$L_n(z)=\sum_{M\in\mathcal{A}_n}z^{\det(M)}.$$
Then, we have $L_2(z)=\frac1z+z$, $L_3(z)=\frac3z + 1+3z$,
$L_4(z)=\frac{12}z+16+ 14z$, 
$$L_5(z)=\frac{86}z + 253  + 90z, \qquad \text{and} \qquad
L_6(z)=\frac2{z^2}+ \frac{1032}z + 5368 + 1032z+ 2z^2.$$
This way, Question 1 reads $L_n'(1)$ where $'=\frac{d}{dz}$.
Credit. These computations are made courtesy of Zeilberger's Maple package, ROBBINS.

REPLY [3 votes]: (Half of an answer to Question 1)
$M \in \mathcal{A}_n \Leftrightarrow w_0M \in \mathcal{A}_n$, where $w_0 := \mathrm{antidiag}(1,1,...,1)$ (this is just a flip across an axis of symmetry), so when $\binom{n}{2}$ is odd (i.e., $n \equiv 2,3 \mod 4$) we have $\sum_{M \in \mathcal{A}_n} \mathrm{det}(M) = 0$ since in that case $\mathrm{det}(w_0) = -1$, as observed in your data.