How can I calculate $\left[{n\atop k}\right]$?

Question

$\left[\begin{array}{c}4\\2\end{array}\right]=11;\left[\begin{array}{c}7\\3\end{array}\right]=1624;$ and $\left[\begin{array}{c}7\\2\end{array}\right]=-1764$

How can I calculate $\left[\begin{array}{c}15\\6\end{array}\right]=?$ or $\left[\begin{array}{c}37\\11\end{array}\right]=?$

What a formula hidden behind these brackets? $\left[\begin{array}{c}n\\k\end{array}\right]$

Answer 1

That notation usually refers to the unsigned Stirling numbers of the first kind, but your ${7\brack 2}=-1764$ indicates that someone is using it for the signed Stirling numbers of the first kind.

The easiest way to compute them is by using the recurrence

$${{n+1}\brack k}=n{n\brack k}+{n\brack{k-1}}$$

for the unsigned numbers, with initial conditions ${0\brack 0}=1$ and ${n\brack 0}={0\brack n}=0$ for $n>0$. The signed version of ${n\brack k}$ is obtained by multiplying the unsigned version by $(-1)^{n-k}$. There is no nice closed form.