A10621 | 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 $mod$ .

The single line of the input contains three integers $n$ , $d$ and $mod$ ( $1<=n<=1000$ , $2<=d<=10$ , $10^{8}<=mod<=10^{9}$ ) — the number of vertices in the tree, the degree of internal vertices and the prime modulo.

Print the number of trees over the modulo $mod$ .