Both errors are caused by overflow. stirlings1 and stirlings2 take arguments of type Int, which is usually Int64, but

• $s(26, 10) \simeq 2^{77} > 2^{63}-1 =$ typemax(Int64),
• $S(26, 10) \simeq 2^{63.5} > 2^{63}-1 =$ typemax(Int64).

#193 addresses this by allowing stirlings1 and stirlings2 to be called with arguments of any subtype of Integer:

julia> stirlings1(big(26), 10)
196928100451110820242880

julia> stirlings2(big(26), 10)
13199555372846848005