Suppose
$$\det\left[\begin{array}{cc}\sum_{k=0}^{n} k & \sum_{k=0}^{n} {}^nC_k k^2 \\ \sum_{k=0}^{n} {}^nC_k & \sum_{k=0}^{n} {}^nC_k 3^k\end{array}\right] = 0$$
holds for some positive integer $n$. Then $\sum_{k=0}^{n} \frac{{}^nC_k}{k+1}$ equals\_\_\_\_