What is the sum of the expression?

$\sum _{n=0}^{100} \left| \left[\left( \begin{array}{c} i+j \\ i \\ \end{array} \right),_{ \begin{array}{c} 0\leq i\leq n\text{, monotonically increasing} \\ 0\leq j\leq n\text{, monotonically increasing} \\ \end{array} } \right]\right|$

Each or the elements of the square matrix is a binomial number. The matrix is enclosed in square brackets above. The vertical bars indicate the computation of the determinant of the square matrix so constructed. For a $3\times 3$ matrix, that matrix would be $\begin{array}{r} 1 & 1 & 1 \\ 1 & 2 &3 \\ 1 & 3 & 6 \\ \end{array}$ .