Very Accurate Approximations for the Factorial Function

We establish the following new Stirling-type approximation formulas for the factorial function n! ≈ √ 2πnne−n √ n+ 1 6 + 1 72n − 31 6480n2 − 139 155520n3 + 9871 6531840n4 and n! ≈ √ 2πnne−n 4 √ n2 + n 3 + 1 18 − 2 405n − 31 9720n2 . Our estimations give much more accurate values for the factorial function than some previously published strong formulas. We also derive new sequences converging to Euler-Mascheroni constant γ very quickly.