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Problem 936

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Peerless Trees

A peerless tree is a tree with no edge between two vertices of the same degree. Let $P(n)$ be the number of peerless trees on $n$ unlabelled vertices.

There are six of these trees on seven unlabelled vertices, $P(7)=6$, shown below.

Define ${\displaystyle S(N)=\sum _{n=3}^{N}P(n)}$. You are given $S(10)=74$.

Find $S(50)$.

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无同树

无同树是指没有连接相同度数顶点的边的树。记$P(n)$为包含$n$个无标记顶点的无同树的数量。

包含七个无标记顶点的无同树有六棵，即$P(7)=6$，如下图所示。

定义${\displaystyle S(N)=\sum _{n=3}^{N}P(n)}$。已知$S(10)=74$。

求$S(50)$。

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