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# Problem 650

## Divisors of Binomial Product

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

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

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

## 二项式系数乘积的因数

$B(5)={5 \choose 0} \times {5 \choose 1} \times {5 \choose 2} \times {5\choose 3}\times {5 \choose 4}\times {5 \choose 5} = 1\times 5\times 10\times 10\times 5\times 1=2500$。