Problem 831

Triple Product

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

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

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：

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

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。