What is the pattern or relation in this table? Here is the table:
$$\begin{array}{c}
0\\  
1\\  
1& 1\\  
3& 2& 3\\  
5& 3& 3& 5\\  
11 &8 &10 &8 &11\\  
21 &13 &14 &14 &13 &21\\  
43 &30 &37 &36 &37 &30 &43\\  
85 &55 &61 &55 &55 &61 &55 &85\\  
171 &116 &140 &140 &146 &140 &140 &116 &171  
\end{array}$$
In this triangle like table, I am unable to find the relation between columnwise.
Say for 1st column, I have found the relation $A(n)= A(n-1)+2A(n-2)$.
For the 2nd column, I have the relation $A(n)= A(n-1)+2A(n-2) + x$ [ where $x$ is $-1$ for even row number and $+1$ for odd].
But for other columns, I have not found the relations yet. Can anyone help me find a generalized equation to generate the table?

Updated:
let $A(n,k)$ be the number in the table with row $n$ and column $k$, where $0\le n$ and $0\le k\le n$.
$$A(n,k)=A(n,n-k)$$
$$A(n,0)=A(n,n)=0$$
The table would look like this: (top left is row $0$ and column $0$)
$$\begin{array}{c}
0\\
0& 0\\  
0& 1& 0\\  
0& 1& 1& 0\\  
0& 3& 2& 3& 0\\  
0& 5& 3& 3& 5& 0\\  
0& 11 &8 &10 &8 &11& 0\\  
0& 21 &13 &14 &14 &13 &21& 0\\  
0& 43 &30 &37 &36 &37 &30 &43& 0\\  
0& 85 &55 &61 &55 &55 &61 &55 &85& 0\\  
0& 171 &116 &140 &140 &146 &140 &140 &116 &171  & 0
\end{array}$$
Then, we have: $$A(n+1,k)+A(n,k)+A(n,k-1)=2^{n-1}$$

Example
$$A(8,3)+A(7,3)+A(7,2)=37+14+13=64=2^6$$
