CF724F.Uniformly Branched Trees

A tree is a connected graph without cycles.

Two trees, consisting of n vertices each, are called isomorphic if there exists a permutation p: {1, ..., n} → {1, ..., n} such that the edge (u, v) is present in the first tree if and only if the edge (p__u, p__v) is present in the second tree.

Vertex of the tree is called internal if its degree is greater than or equal to two.

Count the number of different non-isomorphic trees, consisting of n vertices, such that the degree of each internal vertex is exactly d. Print the answer over the given prime modulo mod.

树是无环的连通图。

由 $n$ 个顶点构成的两棵树被称为同构的，若存在一个置换 $p:\{1,\dots,n\}\to\{1,\dots,n\}$，使得边 $(u,v)$ 在第一棵树中存在当且仅当边 $(p_u,p_v)$ 在第二棵树中存在。

树的一个顶点称为内部顶点，当且仅当它的度数大于等于 $2$。

计算满足以下条件的互不同构的树的数目：该树恰含 $n$ 个顶点，且每个内部顶点的度数恰好为 $d$。将答案对给定的素数模 $mod$ 取模后输出。

The single line of the input contains three integers n, d and mod (1 ≤ n ≤ 1000, 2 ≤ d ≤ 10, 108 ≤ mod ≤ 109) — the number of vertices in the tree, the degree of internal vertices and the prime modulo.

输入仅包含一行，三个整数 $n$、$d$ 和 $mod$（$1 ≤ n ≤ 1000$，$2 ≤ d ≤ 10$，$10^8 ≤ mod ≤ 10^9$）—— 分别表示树的顶点数、内部顶点的度数以及模数（质数）。

Print the number of trees over the modulo mod.

在模 mod 意义下输出树的个数。