Too Many Twos (noch nicht ├╝bersetzt)

Problem 792

We define $\nu_2(n)$ to be the largest integer $r$ such that $2^r$ divides $n$. For example, $\nu_2(24) = 3$.

Define $\displaystyle S(n) = \sum_{k = 1}^n (-2)^k\binom{2k}k$ and $u(n) = \nu_2\Big(3S(n)+4\Big)$.

For example, when $n = 4$ then $S(4) = 980$ and $3S(4) + 4 = 2944 = 2^7 \cdot 23$, hence $u(4) = 7$.
You are also given $u(20) = 24$.

Also define $\displaystyle U(N) = \sum_{n = 1}^N u(n^3)$. You are given $U(5) = 241$.

Find $U(10^4)$.