Let $g(m)$ be the integer defined by the following double sum of products of binomial coefficients:

$$\sum_{j=0}^m\sum_{i = 0}^j (-1)^{j-i}\binom mj \binom ji \binom{j+5+6i}{j+5}.$$

You are given that $g(10) = 127278262644918$.
Its first (most significant) five digits are $12727$.
Find the first ten digits of $g(142857)$ when written in base $7$.