Remainder of Polynomial Division

For positive integers $n$ and $m$ , we define two polynomials $F_{n} (x) = x^{n}$ and $G_{m} (x) = (x - 1)^{m}$ .
We also define a polynomial $R_{n, m} (x)$ as the remainder of the division of $F_{n} (x)$ by $G_{m} (x)$ .
For example, $R_{6, 3} (x) = 15 x^{2} - 24 x + 10$ .

Let $C (n, m, d)$ be the absolute value of the coefficient of the $d$ -th degree term of $R_{n, m} (x)$ .
We can verify that $C (6, 3, 1) = 24$ and $C (100, 10, 4) = 227197811615775$ .

Find $C (10^{13}, 10^{12}, 10^{4}) mod 999999937$ .

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.