Pentatope number

A pentatope 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.

The first few numbers of this kind are :

1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365 (sequence A000332 in OEIS)

A pentatope with side length 5 contains 70 spheres. Each layer represents one of the first five tetrahedral numbers. For example the bottom (green) layer has 35 spheres in total.

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

${n + 3 \choose 4} = \frac{n(n+1)(n+2)(n+3)}{24} = {n^{\overline 4} \over 4!}.$

Two of every three pentatope numbers are also pentagonal numbers. To be precise, the (3k − 2)th pentatope number is always the ((3k² − k)/2)th pentagonal number and the (3k − 1)th pentatope number is always the ((3k² + k)/2)th pentagonal number. The 3kth pentatope number is the generalized pentagonal number obtained by taking the negative index −(3k² + k)/2 in the formula for pentagonal numbers. (These expressions always give integers).

The infinite sum of the reciprocals of all pentatopal numbers is $4 \over 3$. This can be derived using telescoping series.

$\sum_{n=1}^\infty {4! \over {n(n+1)(n+2)(n+3)}} = {4 \over 3}$

Pentatopal numbers can also be represented as the sum of the first n tetrahedral numbers.

Test for pentatope numbers

$\frac{\sqrt{24n+1}+1}{2}$ is triangular number.

then : $8\times\frac{\sqrt{24n+1}+1}{2}+1$ is perfect square.