Suppose $\det \begin{bmatrix} \sum_{k=0}^{n} k & \sum_{k=0}^{n} {}^{n}C_{k}k^{2} \\ \sum_{k=0}^{n} {}^{n}C_{k}k & \sum_{k=0}^{n} {}^{n}C_{k}3^{k} \end{bmatrix} = 0$ holds for some positive integer $n$. The $\sum_{k=0}^{n} \frac{{}^{n}C_{k}}{k+1}$ equals ______

Suppose det ∑ₖ₌₀ⁿ k ∑ₖ₌₀ⁿ ⁿCₖk² ∑ₖ₌₀ⁿ ⁿCₖk ∑ₖ₌₀ⁿ ⁿCₖ3ᵏ = 0 holds for some positive integer n. The ∑ₖ₌₀ⁿ ⁿCₖ/(k+1) equals ______ $\begin{vmatrix} \sum_{k=0}^{n} k & \sum_{k=0}^{n} {}^{n}C_{k}k^{2} \\[4pt] \sum_{k=0}^{n} {}^{n}C_{k}k & \sum_{k=0}^{n} {}^{n}C_{k}3^{k} \end{vmatrix} = 0$ $\text{for some positive integer } n$ $\text{To find: } \sum_{k=0}^{n} \dfrac{{}^{n}C_{k}}{k+1}$ $1.\ \text{Evaluate the four entries}$ $2.\ \text{Set } \det = 0 \Rightarrow n$ $3.\ \text{Find } \sum_{k=0}^{n} \dfrac{{}^{n}C_{k}}{k+1}$ $A = \sum_{k=0}^{n} k = \dfrac{n(n+1)}{2}$ $C = \sum_{k=0}^{n} {}^{n}C_{k}\,k = n\,2^{n-1}$ $B = \sum_{k=0}^{n} {}^{n}C_{k}\,k^2 = \sum_{k=0}^{n} {}^{n}C_{k}\,[k(k-1)+k]$ $= n(n-1)2^{n-2} + n\,2^{n-1} = n(n+1)2^{n-2}$ $D = \sum_{k=0}^{n} {}^{n}C_{k}\,3^k = (1+3)^n = 4^n$ $\det = A D - B C = 0$ $A D = \dfrac{n(n+1)}{2}\cdot 4^n = n(n+1)2^{2n-1}$ $B C = n(n+1)2^{n-2}\cdot n\,2^{n-1} = n^2(n+1)2^{2n-3}$ $n(n+1)2^{2n-3}(4 - n) = 0$ $n = 4 \quad (\text{positive integer})$ $\sum_{k=0}^{4} \dfrac{{}^{4}C_{k}}{k+1} = ?$ $\dfrac{{}^{n}C_{k}}{k+1} = \dfrac{{}^{n+1}C_{k+1}}{n+1}$ $\sum_{k=0}^{n} \dfrac{{}^{n}C_{k}}{k+1} = \dfrac{1}{n+1}\sum_{j=1}^{n+1} {}^{n+1}C_{j} = \dfrac{2^{n+1}-1}{n+1}$ $= \dfrac{2^{5}-1}{5} = \dfrac{31}{5}$ $\boxed{\dfrac{31}{5}}$