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)