Question 173
Determine all nonzero natural numbers $n$ for which the number
$$
S_{n}=1 \cdot C_{2 n}^{1}+2 \cdot C_{2 n}^{2}+3 \cdot C_{2 n}^{3}+\ldots+n \cdot C_{2 n}^{n}
$$
is a perfect square less than $1000000$.
$$
S_{n}=1 \cdot C_{2 n}^{1}+2 \cdot C_{2 n}^{2}+3 \cdot C_{2 n}^{3}+\ldots+n \cdot C_{2 n}^{n}
$$
is a perfect square less than $1000000$.