Pentatope number

EN Texto em inglês Ainda não há artigo no teu idioma: trecho em inglês.

In number theory, a pentatope number (or hypertetrahedral number or triangulo-triangular number) is a number in the fifth cell of any row of Pascal's triangle starting with the 5-term row 1 4 6 4 1, either from left to right or from right to left. It is named because it represents the number of 3-dimensional unit spheres which can be packed into a pentatope (a 4-dimensional tetrahedron) of increasing side lengths. The first few numbers of this kind are: Pentatope numbers belong to the class of figurate numbers, which can be represented as regular, discrete geometric patterns. The formula for the nth pentatope number is represented by the 4th rising factorial of n divided by the factorial of 4: The pentatope numbers can also be represented as binomial coefficients: which is the number of distinct quadruples that can be selected from n + 3 objects, and it is read aloud as "n plus three choose four". Two of every three pentatope numbers are also pentagonal numbers. To be precise, the (3k − 2)th pentatope number is always the ( 3 k 2 − k 2 ) {\displaystyle \left({\tfrac {3k^{2}-k}{2}}\right)} th pentagonal number and the (3k − 1)th pentatope number is always the ( 3 k 2 + k 2 ) {\displaystyle \left({\tfrac {3k^{2}+k}{2}}\right)} th pentagonal number. The (3k)th pentatope number is the generalized pentagonal number obtained by taking the negative index − 3 k 2 + k 2 {\displaystyle -{\tfrac {3k^{2}+k}{2}}} in the formula for pentagonal numbers. (These expressions always give integers). The infinite sum of the reciprocals of all pentatope numbers is ⁠4/3⁠. This can be derived using telescoping series.