Problem Statement

Lah numbers, sometimes referred to as Stirling numbers of the third kind, are coefficients of polynomial expansions expressing rising factorials in terms of falling factorials. Unsigned Lah numbers count the number of ways a set of n elements can be partitioned into k non-empty linearly ordered subsets. Lah numbers are closely related to Stirling numbers of the first & second kinds, and may be derived from them. Lah numbers obey the identities and relations:

Let's start with the solution:

// Lah numbers. Nigel Galloway: January 3rd., 2023
let fact(n:int)=let rec fact=function n when n=0I->1I |n->n*fact(n-1I) in fact(bigint n)
let rec lah=function (_,0)|(0,_)->0I |(n,1)->fact n |(n,g) when n=g->1I |(n,g)->((fact n)*(fact(n-1)))/((fact g)*(fact(g-1)))/(fact(n-g))
for n in {0..12} do (for g in {0..n} do printf $"%A{lah(n,g)} "); printfn ""
printfn $"\n\n%A{seq{for n in 1..99->lah(100,n)}|>Seq.max}"