# Lah-numbers and the coefficients of Laguerre polynomials

Wouter Meeussen w.meeussen.vdmcc at vandemoortele.be
Thu Jan 25 15:10:08 CET 2001

```Lah-numbers and the coefficients of Laguerre polynomials:
a link between A000142, A001563, A001286, A001809, A001754, A001810, A001755, A001811, A001777, ...

folding an anonymous function (1/#1- #2)& over a list of elements {a[1],a[2],a[3]} gives :

FoldList[(1/#1-#2)&,1,Array[a,3]]//Together
{1, 1 - a[1], (-1 + a[2] - a[1]*a[2])/(-1 + a[1]), (1 - a[1] - a[3] + a[2]*a[3] - a[1]*a[2]*a[3])/(1 - a[2] + a[1]*a[2])}

this has structure   A, B/A , C/B, D/C ..  ; the numerators containing all the info.

Now, change the function into (1/#1- x #2)& and we get numerators that are polynomes in x:
1,
1 - x,
-1 + 2*x - 2*x^2,
1 - 4*x + 6*x^2 - 6*x^3,
-1 + 6*x - 18*x^2 + 24*x^3 - 24*x^4,
1 - 9*x + 36*x^2 - 96*x^3 + 120*x^4 - 120*x^5
....
and a triangle of coefficients like:
{1}
{1, -1}
{-1, 2, -2}
{1, -4, 6, -6}
{-1, 6, -18, 24, -24}
{1, -9, 36, -96, 120, -120}
{-1, 12, -72, 240, -600, 720, -720}
{1, -16, 120, -600, 1800, -4320, 5040, -5040}
{-1, 20, -200, 1200, -5400, 15120, -35280, 40320, -40320}
{1, -25, 300, -2400, 12600, -52920, 141120, -322560, 362880, -362880}
{-1, 30, -450, 4200, -29400, 141120, -564480, 1451520, -3265920, 3628800, -3628800}
{1, -36, 630, -7350, 58800, -376320, 1693440, -6531840, 16329600, -36288000, 39916800, -39916800}
{-1, 42, -882, 11760, -117600, 846720, -5080320, 21772800, -81648000, 199584000, -439084800, 479001600, -479001600}

the diagonals are:

A000142=Table[(n)!,{n,1,12}]
{1, 2, 6, 24, 120, 720, ...

A001563=Table[n(n)!,{n,1,12}]
{1, 4, 18, 96, 600, 4320, 35280, 322560, 3265920, 36288000, 439084800, 5748019200}

A001286=Table[(-1)^n * n * (n+1)! /2,{n,1,12}]
{-1, 6, -36, 240, -1800, 15120, -141120, 1451520, -16329600, 199584000, -2634508800, 37362124800}

A001809=Table[n!*n*(n-1)/4,{n,1,12}] ; coefficients of Laguerre polynomials
{0, 1, 9, 72, 600, 5400, 52920, 564480, 6531840, 81648000, 1097712000, 15807052800}

A001754=Table[n! Binomial[n-1,2]/6,{n,1,12}] ; Name: Lah numbers:
{0, 0, 1, 12, 120, 1200, 12600, 141120, 1693440, 21772800, 299376000, 4390848000}

A001810=Table[n!*n*(n-1)(n-2)/36,{n,1,12}] ; coefficients of Laguerre polynomials
{0, 0, 1, 16, 200, 2400, 29400, 376320, 5080320, 72576000, 1097712000, 17563392000}

A001755=Table[n! Binomial[n-1,3]/4!,{n,1,12}] ; Name: Lah numbers:
{0, 0, 0, 1, 20, 300, 4200, 58800, 846720, 12700800, 199584000, 3293136000}

A001811=Table[n!*n*(n-1)(n-2)(n-3)/(4!)^2,{n,1,12}]
{0, 0, 0, 1, 25, 450, 7350, 117600, 1905120, 31752000, 548856000, 9879408000}

A001777=Table[n! Binomial[n-1,4]/5!,{n,1,12}]
{0, 0, 0, 0, 1, 30, 630, 11760, 211680, 3810240, 69854400, 1317254400}

two items remain:
give a general formula for the coefficient list above, (possibly different for even||odd n)
show the link between this folding-operation, the Lah-numbers and the (coefficients of "some??" Laguerre-polynomials)
Any suggestions for improvement?

Wouter Meeussen
tel +32 (0)51 33 21 24
fax +32 (0)51 33 21 75
wouter.meeussen at vandemoortele.com

```