Computation of Unsigned Stirling Numbers of the First Kind

The unsigned Stirling numbers of the first kind, \(\begin{bmatrix} n\\ m \end{bmatrix}\), represent the number of ways to arrange \(n\) distinct elements into \(m\) circular permutations (cycles). Given two integers \(n\) and \(k\), compute \(\begin{bmatrix} i\\ k \end{bmatrix}\) modulo \(167772161\) (where \(167772161 = 2^{25}\times 5+1\), a prime number) for all integers \(i \in [0, n]\).

The recurrence relation is given by:

[ \begin{bmatrix} n \ k \end{bmatrix} = \begin{bmatrix} n-1 \ k-1 \end{bmatrix} + (n-1) \begin{bmatrix} n-1 \ k \end{bmatrix} ]

with the initial conditions \(\begin{bmatrix} 0 \\ 0 \end{bmatrix} = 1\) and \(\begin{bmatrix} n \\ k \end{bmatrix} = 0\) if \(n < k\) or \(k 0\), we have \(\begin{bmatrix} n \\ 0 \end{bmatrix} = 0\).

inputFormat

The input consists of two space-separated integers:

1. \(n\): the maximum number of elements.
2. \(k\): the fixed number of cycles.

\(n\) and \(k\) may be provided in one line.

outputFormat

Output \(n+1\) numbers separated by spaces. The \(i\)th number (where \(i\) ranges from \(0\) to \(n\)) is the value of \(\begin{bmatrix} i \\ k \end{bmatrix}\) modulo \(167772161\).

sample

5 3

0 0 0 1 6 35