Sage code:

    values = {}
    def A(n,k):
        if (n,k) in values:
            return values[(n,k)]
        if n==0:
            if k==0:
                result = 1
            else:
                result = 0
        else:
            result =  (n + 1 - k)**2 * A(n-1, k-1)
            result += 2*(n*(n+1)-k**2) * A(n-1, k)
            result += (n + 1 + k)**2 * A(n-1, k+1)
        values[(n,k)]=result
        return result
    
    for n in range(5):
        print [A(n,k) for k in range(-n, n+1)]

Output:

    [1]
    [1, 4, 1]
    [1, 26, 66, 26, 1]
    [1, 120, 1191, 2416, 1191, 120, 1]
    [1, 502, 14608, 88234, 156190, 88234, 14608, 502, 1]

These are famous numbers to combinatorists.  So it looks like you're getting Eulerian numbers for odd n, with k shifted.  See A008292.  Note that it's easy to see that A(n,k) = A(n,-k) by induction on n, even if you didn't know that.