Strategies on the evaluation of binomial coefficients for all integers

Binomial coefficients are used in many fields such as computational and applied mathematics, statistics and probability, theoretical physics and chemistry. For accurate numerical results, the correct calculation of these coefficients is very important. We present some new recurrence relationships and numerical methods for the evaluation of binomial coefficients for negative integers. For this purpose, we give some comparisons of the outputs for different computer programming languages in case of negative integers, and also we wrote two new algorithms for computations.