On some methods of deriving geometric expectation

Formulas presented for the calculation of ∑ _j=1 ⁿ j^k (n, k ∈ N do not have a closed form; they are in the form of recursive or complex formulas. Here an attempt is made to present a simple formula in which it is only necessary to compute the numerical coefficients in a recursive form, and the coefficients in turn follow a simple pattern (almost similar to Pascal's Triangle). Although the pattern for calculating numerical coefficients based on forming a table is easy, non-recursive formulas are presented to determine the numerical coefficients.