Here are three reasons for this to be false :

• When $k=1, n=2, a_1=1, b_1=0$ I get $(j+1)(j+2)$ on the LHS and $1 + \binom{3+j}{3}$ on the RHS.
• The (ordinary) generating function of the LHS (with respect to $j$) is $$ \sum_{m=1}^{k+1} \frac{1}{(1-X)^{a_{m-1} - a_m + k + 1}} $$ which is certainly not (in general) independent from the sequence $a_1 , ... , a_{k+1}$.
• The LHS is polynomial in $j$ of degree $k + \max_{1 \leq m \leq k+1} (a_{m-1} - a_m) $ which may differ from $k + \max_{1 \leq m \leq k+1} (b_{m-1} - b_m) $

Here are three reasons for this to be false :

• When $k=1, n=2, a_1=1, b_1=0$ I get $(j+1)(j+2)$ on the LHS and $j+1 + \binom{3+j}{3}$ on the RHS.
• The (ordinary) generating function of the LHS (with respect to $j$) is $$ \sum_{m=1}^{k+1} \frac{1}{(1-X)^{a_{m-1} - a_m + k + 1}} $$ which is certainly not (in general) independent from the sequence $a_1 , ... , a_{k+1}$.
• The LHS is polynomial in $j$ of degree $k + \max_{1 \leq m \leq k+1} (a_{m-1} - a_m) $ which may differ from $k + \max_{1 \leq m \leq k+1} (b_{m-1} - b_m) $