?PoissonDistribution

PDF[PoissonDistribution[λ],n]

DiscretePlot[Table[PDF[PoissonDistribution[λ],k],{λ,{5,10,20}}]//Evaluate,{k,0,30},PlotRange->All,PlotMarkers->Automatic,Frame->True]

(* Large n limit. Looks like a gaussian! *)

DiscretePlot[PDF[PoissonDistribution[100],n],{n,0,200},PlotRange->All]

PDF[PoissonDistribution[λ],n]

(* Stirling’s formula *) 
Series[n!,{n,∞,0}]//Normal//PowerExpand//FullSimplify

(* Expand for large λ and n *)

ss=Series[(((λ^n) Exp[-λ])/(n!))/.n->λ(1+δ),{λ,∞,-1}]/.Floor[x_]->0//Normal//PowerExpand//FullSimplify

(* Write the (((1+δ))^(-(1/2)-(1+δ) λ)) term as Exp[Log[(((1+δ))^(-(1/2)-(1+δ) λ))]] and expand *)

ss1=Series[Log[(((1+δ))^(-(1/2)-(1+δ) λ))],{δ,0,2},{λ,∞,-1}]//FullSimplify//Normal

p=(((E^(δ λ))Exp[ss1])/((Sqrt[2 π]) (Sqrt[λ])))/.δ->(n-λ)/λ

Integrate[p,{n,-∞,∞},GenerateConditions->False]//PowerExpand