Pentatope Number

A Figurate Number which is given by

$${\it Ptop}_n={\textstyle{1\over 4}}{\it Te}_n(n+3)={\textstyle{1\over 24}} n(n+1)(n+2)(n+3),$$

where ${\it Te}_n$ is the $n$th Tetrahedral Number. The first few pentatope numbers are 1, 5, 15, 35, 70, 126, ... (Sloane's A000332). The Generating Function for the pentatope numbers is

$${x\over(1-x)^5}=x+5x^2+15x^3+35x^4+\ldots.$$

See also Figurate Number, Tetrahedral Number

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 55-57, 1996.