Pentatope number

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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 Template:OEIS

Image:Pentatope of 70 spheres animation.gif 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: <math>{{n + 3} \choose 4} = n(n+1)(n+2)(n+3) / 24.</math>

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

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fr:Nombre pentatopique
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