7 Express the polynomial in $x$, $f(x)=1-x+x^{2}-x^{3}+\cdots-x^{19}+$ $x^{20}$, as a polynomial in $y$, $g(y)=a_{0}+a_{1} y+a_{2} y^{2}+\cdots+a_{19} y^{19}+$ $a_{20} y^{20}$, where $y=x-4$. Then $a_{0}+a_{1}+\cdots+a_{20}=$ $\qquad$ .
From the given, we know that the terms in the series of $f(x)$ form a geometric sequence with the first term being 1 and the common ratio being $-x$. By the formula for the sum of a geometric series, we get $f(x)=\frac{(-x)^{21}-1}{-x-1}=$ $\frac{x^{21}+1}{x+1}$.

Let $x=y+4$, then $g(y)=\frac{(y+4)^{21}+1}{y+5}$. Taking $y=1$, we have $a_{0}+a_{1}+$ $a_{2}+\cdots+a_{20}=g(1)=\frac{5^{21}+1}{6}$.