# 201 : 唯一の和を持つ部分集合

数の集合$$A$$について,$$sum(A)$$で$$A$$の要素の和を表す. 集合$$B = \{1,3,6,8,10,11\}$$を考えよう. $$B$$の$$3$$要素の部分集合は$$20$$個あり, それぞれ和は以下になる.

$$
\begin{aligned}
&sum(\{1,3,6\}) = 10, \\
&sum(\{1,3,8\}) = 12, \\
&sum(\{1,3,10\}) = 14, \\
&sum(\{1,3,11\}) = 15, \\
&sum(\{1,6,8\}) = 15, \\
&sum(\{1,6,10\}) = 17, \\
&sum(\{1,6,11\}) = 18, \\
&sum(\{1,8,10\}) = 19, \\
&sum(\{1,8,11\}) = 20, \\
&sum(\{1,10,11\}) = 22, \\
&sum(\{3,6,8\}) = 17, \\
&sum(\{3,6,10\}) = 19, \\
&sum(\{3,6,11\}) = 20, \\
&sum(\{3,8,10\}) = 21, \\
&sum(\{3,8,11\}) = 22, \\
&sum(\{3,10,11\}) = 24, \\
&sum(\{6,8,10\}) = 24, \\
&sum(\{6,8,11\}) = 25, \\
&sum(\{6,10,11\}) = 27, \\
&sum(\{8,10,11\}) = 29. \\
\end{aligned}
$$

これらの和は$$1$$つしか現れない場合もそうでない場合もある. 集合$$A$$について, $$U(A,k)$$で, $$A$$の$$k$$要素の集合全体について和を取ったときに$$1$$回のみ現れる和の集合を表す. 上の例をとると, $$U(B,3) = \{10,12,14,18,21,25,27,29\}$$であり, $$sum(U(B,3)) = 156$$となる.

今, $$100$$個の要素を持つ集合 $$S = \{1^2, 2^2, ..., 100^2\}$$を考える. $$S$$の$$50$$要素の部分集合は$$100891344545564193334812497256$$個ある.

$$50$$要素の部分集合の和の中で$$1$$回のみ現れる和の集合の総和を求めよ. すなわち, $$sum(U(S,50))$$を求めよ.
