Find number of positive integer solutions to the equation $a+b+c=15$ where a,b are even and c is odd This is equivalent to:
Coefficient of $x^{15}$ in $(1+x^2+x^4+\cdots+x^{14})^2(x+x^3+\cdots+x^{15})$
Or
Coefficient of $x^{14}$ in $(1+x^2+x^4+\cdots+x^{14})^3$
Is this approach correct?

It's already been pointed out that the OP has an error in that $a,b,c$ are supposed to be positive, not non-negative.
But I would like to point out that the algebra is a little simpler if we use  the constraints $a,b,c \ge 1$ without taking into account our knowledge that $a,b,c \le 15$.  This gives the generating function
$$\begin{align} 
f(x) &= (x^2 + x^4 + x^6 + \dots)^2 (x + x^3 + x^6 + \dots) \\
&= x^5 (1 + x^2 + x^4 + \dots)^3 \\
&= x^5 \left( \frac {1}{1-x^2} \right)^3 \\
&= x^5 (1-x^2)^{-3} \\
&= x^5 \sum_{i=0}^{\infty} \binom{3+i-1}{i} x^{2i}
\end{align}$$
From the last equation we see that the coefficient of $x^{15}$ is $\binom{7}{5} = 21$.
