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?

You say "positive" rather than "non-negative", so it is

*

*the coefficient of $x^{15}$ in $(x^2+x^4+\cdots+x^{14})^2(x+x^3+\cdots+x^{15})$
which is

*

*the coefficient of $x^{15}$ in $(x^2+x^4+\cdots+x^{12})^2(x+x^3+\cdots+x^{11})$

*the coefficient of $x^{16}$ in $(x^2+x^4+\cdots+x^{12})^3$

*the coefficient of $x^{10}$ in $(1+x^2+x^4+\cdots+x^{10})^3$

*the coefficient of $x^{5}$ in $(1+x+x^2+\cdots+x^{5})^3$
which is ${7 \choose 2}=21$
That is short enough to list as a check:
2 + 2 + 11
2 + 4 + 9
2 + 6 + 7
2 + 8 + 5
2 + 10 + 3
2 + 12 + 1
4 + 2 + 9
4 + 4 + 7
4 + 6 + 5
4 + 8 + 3
4 + 10 + 1
6 + 2 + 7
6 + 4 + 5
6 + 6 + 3
6 + 8 + 1
8 + 2 + 5
8 + 4 + 3
8 + 6 + 1
10 + 2 + 3
10 + 4 + 1
12 + 2 + 1

