Problems with Stirling numbers

[scheinerman]

Stirling numbers of the second kind are always nonnegative and yet:

julia> stirlings2(26,10)
-5247188700862703611

The same issue plagues stirlings1. We have

julia> stirlings1(26,10)
9107464261356742080

but the correct value is 196928100451110820242880.

[FedericoStra]

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