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A Pentatope Number is a figurate number (a number that can be represented by a regular geometric arrangement of equally spaced points) given by

where Tn is the nth tetrahedral number.

The first few pentatope numbers are 1, 5, 15, 35, 70, 126, ... and they are found in the fifth diagonal of Pascal's triangle:

 
 

 

 

 

 

 

 

 

 1

 

 

 

 

 

 

 

 

  Row 0
 

 

 

 

 

 

 

 1

 

 1

 

 

 

 

 

 

 

  Row 1
 

 

 

 

 

 

 1

 

 2

 

 1

 

 

 

 

 

 

  Row 2
 

 

 

 

 

  1

 

 3

 

 3

 

 1

 

 

 

 

 

  Row 3
 

 

 

 

 1

 

 4

 

 6

 

 4

 

 1

 

 

 

 

  Row 4
 

 

 

 1

 

 5

 

10

 

10

 

 5

 

 1

 

 

 

  Row 5
 

 

 1

 

 6

 

15

 

20

 

15

 

 6

 

 1

 

 

  Row 6
 

 1

 

 7

 

21

 

35

 

35

 

21

 

 7

 

 1

 

  Row 7
 1

 

 8

 

28

 

56

 

70

 

56

 

28

 

 8

 

 1

  Row 8

References

  1. Eric's Treasure Trove of Mathematics: Pentatope Number
  2. Eric's Treasure Trove of Mathematics: Figurate Number


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Math Forum * * * * 2 May 1998