Pattern in the sequence np associated with a new use of Pascal’s triangle [1], which provides p+1 terms composed of known numbers whose coefficients are the first p+1 elements of the Newton’s binomial (a+b)(p+k-1) | k ≥ 1.
This pattern says that the alternating addition and subtraction of the set of elements taken from (n+p)p to np, whose coefficients belong to Newton's binomial (a+b)p, has as a result p!. The same operation with the terms from (n+p+k)p | k ≥ 1 to np using the coefficients of the binomial (a+b)(p+k), invariably results in zero. And if the terms are taken from (n+i)p| 0 ≤ i < p to np using the coefficients of the binomial (a+b)i, the result is a positive number whose value depends on the numbers taken from the sequence.
2020 Mathematics Subject Classifications: 40G99, 65Q99, 11B99, 15B36.