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