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As mentioned by blochwave, your code can't benefit from compilation. To speed up your code, take Rolf Mertig's advise may be the best. Making use of the Listable attribute of those arithmetical function will also help a little: (* Tested under n = 1000, dual-core laptop *) pk1 = Table[ Sum[Binomial[a, i]*StirlingS2[i, r]*(n - c)^(-i), {i, r, a}], {r, 1, ...

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