How to solve this multivariate recursion?:

$$a(m, n, k) = 2a(m-1, n-1, k-1) + a(m-1, n-1, k) + a(m-1, n, k-1) + a(m, n-1, k-1)$$

where $m, n, k$ are positive integers.

Edit:

$a(0,0,0) = a(1,0,0) = a(0,1,0) = a(0,0,1) = 1$

$a(m,0,0) = a(0,m,0) = a(0,0,m) = 0, m > 1$

$a(m,n,k) = a(m,k,n) = a(n,m,k) = a(n,k,m) = a(k,m,n) = a(k,n,m)$