Pentatope number

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 OEIS|id=A000332

[
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 "n"th pentatopic number is::n + 3} choose 4} = frac{n(n+1)(n+2)(n+3)}{24}.

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

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