  Let $A(n,r;3)$ be the total weight of the alternating sign matrices of order
$n$ whose sole `1' of the first row is at the $r^{th}$ column and the weight of
an individual matrix is $3^k$ if it has $k$ entries equal to -1. Define the
sequence of the generating functions $G_n(t)=\sum_{r=1}^n A(n,r;3)t^{r-1}$.
  Results of two different kind are obtained. On the one hand I made the
explicit expression for the even subsequence $G_{2\nu}(t)$ in terms of two
linear homogeneous second order recurrence in $\nu$ (Theorem 1). On the other
hand I brought to light the nice connection between the neighbouring functions
$G_{2\nu+1}(t)$ and $G_{2\nu}(t)$ (Theorem 2).
  The 3-enumeration $A(n;3) \equiv G_n(1)$ which was found by Kuperberg is
reproduced as well.
