Number pentatopic

A number pentatopic is a number of the downward diagonal fifth of the Triangle of Pascal. The first numbers of this kind are 1,5,15,35,70, and 126.

The numbers pentatopic are illustrated numbers. They can ideally be represented in dimension 4 by a Polytope made up of a stacking of Tétraèdre S regular.

The number pentatopic of row N is thus the sum of tetrahedral N first numbers

$P\_4\; (N)\; =\; \backslash \; sum\; \_\; \{k=1\}\; ^\; \{N\}\; P\_3\; (N)\; =\; \backslash \; sum\; \_\; \{k=1\}\; ^\; \{N\}\; \{N\; +2\; \backslash \; choose\; 3\}$

One thus obtains the formula

$P\_4\; (N)\; =\; \{N\; +3\; \backslash \; choose\; 4\}$

It is thus not surprising to meet them in the fifth diagonal of the triangle of Pascal.

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