<s>[INST] sum of coefficient of all even power of $x$ 
The sum of all Coefficient of even power of $x$ in
$(1-x+x^2-x^3+\cdots +x^{2n})(1+x+x^2+x^3+\cdots +x^{2n}),n\in \mathbb{N}$

what i try
for $n=1,$ we have
$(1-x+x^2)(1+x+x^2)=(1+1)-(1)+(1+1)=3$
for $n=2,$ we have
$(1-x+x^2-x^3+x^4)(1+x+x^2+x^3+x^4)$
$=(1+1+1)-(1+1)+(1+1+1)-(1+1)+(1+1+1)=5$
so in this way , get  sum of coefficient in original expression is $2n+1$
but How do i solve it without substituting value of $n$, Help me
 [/INST] These are two standard geometric series,
one with ratio $-x$,
the other with ratio $x$.
Their product is
$(1-(-x)^{2n+1})(1-x^{2n+1})/((1+x)(1-x))\\
=(1+x^{2n+1})(1-x^{2n+1})/(1-x^2)\\
=(1-x^{4n+2})/(1-x^2)
$
At $x=1$, applying Hoppy,
we get
$(4n+2)x^{4n+1}/(2x)
=(2n+1)x^{4n}
\to 2n+1
$.
</s>