Starting at zero, a kangaroo hops along the real number line in the positive direction. Each successive hop takes the kangaroo forward a uniformly random distance between $0$ and $1$. Let $H(n)$ be the expected number of hops needed for the kangaroo to pass $n$ on the real line.

If we write $\alpha =H(1)$, then for all positive integers $n$, $H(n)$ can be expressed as a polynomial function of $\alpha$ with rational coefficients. For example $H(3)={\alpha }^3-2{\alpha }^2+\frac{1}{2}\alpha$. Define $S(n)$ to be the sum of all integer coefficients in this polynomial form of $H(n)$. Therefore $S(1)=1$ and $S(3)=1+(-2)=-1$.
You are also given ${\displaystyle \sum _{n=1}^{10}S(n)=43}$.
Find ${\displaystyle \sum _{n=1}^{{10}^{18}}S(n)}$. Give your answer modulo ${10}^9+7$.