Lah number

A coefficient in the expansion

$$( - x ) ^ {( n ) } = \sum _ {k = 0 } ^ { n } L _ {n,k } x ^ {( k ) } ,$$

where

$$x ^ {( k ) } = x ( x - 1 ) \dots ( x - k + 1 ) , \quad k \geq 1,$$

$$x ^ {( 0 ) } = 1,$$

are the falling factorials.

Replacing $x$ by $- x$, it follows that

$$x ^ {( n ) } = \sum _ {k = 0 } ^ { n } L _ {n,k } ( - x ) ^ {( k ) } .$$

The Lah numbers are given explicitly by

$$L _ {n,k } = ( - 1 ) ^ {n} \left ( \begin{array}{c} {n - 1 } \\ {k - 1 } \end{array} \right ) { \frac{n! }{k! } } , \quad n \geq k \geq 1,$$

$$L _ {0,0 } = 1, \quad L _ {n,0 } = 0, n \geq 1,$$

and they are tabulated in [a1] for $1 \leq k \leq n \leq 10$.

The numbers satisfy the recurrence relation

$$L _ {n + 1,k } = - ( n + k ) L _ {n,k } - L _ {n,k - 1 } ,$$

and have the generating function

$${ \mathop{\rm exp} } ( ut ( 1 - t ) ^ {- 1 } ) = \sum _ {n = 0 } ^ \infty \sum _ {k = 0 } ^ { n } ( - 1 ) ^ {n} { \frac{L _ {n,k } u ^ {k} t ^ {n} }{n! } } .$$

They are related to Stirling numbers of the first and second kinds (cf. Combinatorial analysis), and to Bell polynomials (cf. Bell polynomial) by

$$L _ {n,k } = \sum ( - 1 ) ^ {r} s ( n,r ) S ( r,k ) =$$

$$= ( - 1 ) ^ {n} B _ {n,k } ( 1!, \dots, ( n - k + 1 ) ! ) .$$

See also [a4] for a connection with Laguerre polynomials.

If $a _ {n}$ and $b _ {n}$ are sequences, then

$$a _ {n} = \sum _ { k } L _ {n,k } b _ {k} \iff b _ {n} = \sum _ { k } L _ {n,k } a _ {k} .$$

References

[a1]	L. Comtet, "Advanced combinatorics" , Reidel (1974)
[a2]	I. Lah, "Eine neue Art von Zahlen, ihre Eigenschaften und Anwendung in der mathematischen Statistik" Mitteil. Math. Statist. , 7 (1955) pp. 203–216
[a3]	J. Riordan, "Combinatorial analysis" , Wiley (1958)
[a4]	S. Roman, "The umbral calculus" , Acad. Press (1984)