7. The sum of the coefficients of the terms with integer powers of $x$ in the expansion of $(\sqrt{x}+2)^{2 n+1}$ is

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$(\sqrt{x}+2)^{2 n+1}=\sum_{k=0}^{2 n+1} C_{2 n+1}^{k}(\sqrt{x})^{2 n+1-k} \cdot 2^{k}$, thus $k$ is odd.
Let $A=C_{2 n+1}^{1} \cdot 2+C_{2 n+1}^{3} \cdot 2^{3}+\cdots+C_{2 n+1}^{2 n+1} \cdot 2^{2 n+1}$,
$B=C_{2 n+1}^{0} \cdot 2^{0}+C_{2 n+1}^{2} \cdot 2^{2}+\cdots+C_{2 n+1}^{2 n+1} \cdot 2^{2 n}$.
Then $\left\{\begin{array}{l}B+A=(1+2)^{2 n+1}, \\ B-A=(1-2)^{2 n+1}\end{array} \Rightarrow 2 A=3^{2 n+1}+1 \Rightarrow A=\frac{3^{2 n+1}+1}{2}\right.$.