Special case coefficient sum in multinomial equation

Question

I need to find the sum of coefficients of $x^{c}$ in the general equation

$(1+x)^{a_1}(1+x^2)^{a_2}...(1+x^m)^{a_m}$, where $c$ is a multiple of $m+1$.

For example in $(1+x)^2(1+x^2)$ the coefficient of $x^0$ plus the coefficient of $x^3$ is $3$.