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Mathematics
Problem
Suppose
det
[
∑
k
=
0
n
k
∑
k
=
0
n
n
C
k
k
2
∑
k
=
0
n
n
C
k
k
∑
k
=
0
n
n
C
k
3
k
]
=
0
\det \begin{bmatrix} \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{bmatrix} = 0
det
[
∑
k
=
0
n
k
∑
k
=
0
n
n
C
k
k
∑
k
=
0
n
n
C
k
k
2
∑
k
=
0
n
n
C
k
3
k
]
=
0
holds for some positive integer
n
n
n
. Then
∑
k
=
0
n
n
C
k
k
+
1
\sum_{k=0}^{n} \frac{{}^{n}C_{k}}{k+1}
∑
k
=
0
n
k
+
1
n
C
k
equals __.
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