7. Express the polynomial in $x$
$$
f(x)=1-x+x^{2}-x^{3}+\cdots-x^{19}+x^{20}
$$

as a polynomial in $y$
$$
g(y)=a_{0}+a_{1} y+\cdots+a_{20} y^{20},
$$

where $y=x-4$. Then $a_{0}+a_{1}+\cdots+a_{20}=$
$\qquad$
$=.7 . \frac{5^{21}+1}{6}$.
From the problem, we know that the terms in the expression of $f(x)$ form a geometric sequence with the first term $1$ and the common ratio $-x$.
By the sum formula of a geometric sequence, we have
$$
f(x)=\frac{(-x)^{21}-1}{-x-1}=\frac{x^{21}+1}{x+1} \text {. }
$$

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