Let $A = (1+x+\cdots x^{27})$ and $B=(1+x+\cdots + x^{14})$. Then the desired result is the coefficient of $x^{28}$ in $AB^2$. We note that the degree of every term in the expansion of $B^2$ ranges from $0$ to $28$. For each of these terms except $x^0=1$, there is one and only one term in $A$ which can multiply it to generate a final term of $x^{28}$. Therefore, the desired result equals the number of unconsolidated terms, excluding $x^0=1$, in the expanding of $B^2$ which is $15^2-1=\boxed{224}$.