I used generating functions to calculate the number of ways to divide a group of $n$ people into groups A, B, C and then line them up in the groups, where A must contain an odd number of people and B an even number of people. The exponential generating functions were $$ A(x)=\frac{e^x-e^{-x}}{2}-1, \qquad B(x)=\frac{e^x+e^{-x}}{2}, \qquad C(x)=e^x. $$ By the product rule, the number of ways to put the people in this arrangement is $F(x)=A(x)B(x)C(x)$. Thus $$ F(x)=\frac{1}{4}\sum_{n\ge0}(3^n-2^{n+1}-(-1)^n)\frac{x^n}{n!}+\frac{1}{2}, $$ which gives me the solution: $$ f_0=0, \qquad f_{n\ge1}=\frac{3^n-2^{n+1}-(-1)^n}{4}. $$
Is there a way to get $f_{n\ge1}$?
The problem is that $\displaystyle A(x) = \frac{e^x-e^{-x}}2$, not $\displaystyle A(x) = \frac{e^x-e^{-x}}2 - 1$. (You can tell because the $0$th coefficient, which is the number of ways to line $0$ people up in a group of odd size, should be $0$, not $-1$.) That probably fixes your issue with the coherence of the end formula.