I am doing a counting problem and I ended up with the following equations.
$$ D_n = \sum_{i=1}^{n-1} (D_{n-i}+v_{n-i}+v_i\cdot 2^{n-i-1}),\quad D_1=0, D_2=0, D_3=2, D_4=9. $$ $$ v_n = v_{n-1} + C_{n-2} + D_{n-1} ,\quad v_1 = 0, v_2 = 1, v_3 = 2. $$ $$ C_n = n^{\textrm {th}}\quad \textrm {Catalan Number}. \quad C_1 = 1, C_2 = 2, C_3 = 5, C_4 = 14.$$
I want a formula which will generate the sequence of $v_n$?. Please guide me.
The only thing I have done solving this is taking the difference of $D_n$-$D_n-1$, which is:
$D_n= 2(D_{n-1} + v_{n-1}) + \sum_{i=1}^{n-2} v_i\cdot 2^{n-i-2}$
and replacing $D_{n-1} + v_{n-1}$ with $v_{n} - C_{n-2}$ but it didn't work.