Problem 831

Triple Product

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 $7$.

三个二项式系数的乘积

记$g(m)$为如下表达式（对一系列二项式系数的乘积进行双重求和）所定义的整数：

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

已知$g(10) = 127278262644918$。

它的前五位数字是$12727$。

求$g(142857)$的$7$进制表示的前十位数字。