Divisors of Binomial Product

Let $B (n) = \prod_{k = 0}^{n} (\binom{n}{k})$ , a product of binomial coefficients.
For example, $B (5) = (\binom{5}{0}) \times (\binom{5}{1}) \times (\binom{5}{2}) \times (\binom{5}{3}) \times (\binom{5}{4}) \times (\binom{5}{5}) = 1 \times 5 \times 10 \times 10 \times 5 \times 1 = 2500$ .

Let $D (n) = \sum_{d | B (n)} d$ , the sum of the divisors of $B (n)$ .
For example, the divisors of B(5) are 1, 2, 4, 5, 10, 20, 25, 50, 100, 125, 250, 500, 625, 1250 and 2500,
so D(5) = 1 + 2 + 4 + 5 + 10 + 20 + 25 + 50 + 100 + 125 + 250 + 500 + 625 + 1250 + 2500 = 5467.

Let $S (n) = \sum_{k = 1}^{n} D (k)$ .
You are given $S (5) = 5736$ , $S (10) = 141740594713218418$ and $S (100)$ mod $1 000 000 007 = 332792866$ .

Find $S (20 000)$ mod $1 000 000 007$ .

Write-up coming later

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