Squarefree Binomial Coefficients

The binomial coefficients $C_n^k$ can be arranged in triangular form, Pascal's triangle, like this:

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1

.........

It can be seen that the first eight rows of Pascal's triangle contain twelve distinct numbers: $1, 2, 3, 4, 5, 6, 7, 10, 15, 20, 21$ and $35$.

A positive integer $n$ is called squarefree if no square of a prime divides $n$. Of the twelve distinct numbers in the first eight rows of Pascal's triangle, all except $4$ and $20$ are squarefree. The sum of the distinct squarefree numbers in the first eight rows is $105$.

Find the sum of the distinct squarefree numbers in the first $51$ rows of Pascal's triangle.