Number of spanning arborescences with a specific root in a directed graph
It seems that Kirchhoff's theorem can be used:

Kirchhoff's theorem can be modified to count the number of oriented spanning trees in directed multigraphs. The matrix Q is constructed as follows:
The entry i, j for distinct i and j equals −m, where m is the number of edges from i to j;
The entry i, i equals the indegree of i minus the number of loops at i.
The number of oriented spanning trees rooted at a vertex i is the determinant of the matrix gotten by removing the i-th row and column of Q.