User:A legend/6-simplex numbers

A 5-simplex number is a number in the sixth cell of any row of Pascal's triangle starting with the 6-term row


 * 1 5 10 10 5 1

either from left to right or from right to left.

The first few numbers of this kind are :


 * 1, 6, 21, 56, 126, 252

5-simplex numbers belong in the class of figurate numbers, which can be represented as regular, discrete geometric patterns. The formula for the nth 5-simplex number is:


 * $$\frac{n(n+1)(n+2)(n+3)(n+4)}{120} = {n^{\overline 5} \over 5!}.$$

5-simplex numbers can also be represented as the sum of the first n pentatope numbers.

Category:Figurate numbers