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Diagonals

The diagonals of Pascal's triangle contain the figurate numbers of simplices:

The symmetry of the triangle implies that the n d-dimensional number is equal to the d n -dimensional number.

An alternative formula that does not involve recursion is P d (n) = 1 d ! ∏ k = 0 d − 1 (n + k) = n (d) d ! = (n + d − 1 d), {\displaystyle P_{d}(n)={\frac {1}{d!}}\prod _{k=0}^{d-1}(n+k)={n^{(d)} \over d!}={\binom {n+d-1}{d}},} where n is the rising factorial.

The geometric meaning of a function P is: P (1) = 1 for all d. Construct a d - dimensional triangle (a 3-dimensional triangle is a tetrahedron) by placing additional dots below an initial dot, corresponding to P (1) = 1. Place these dots in a manner analogous to the placement of numbers in Pascal's triangle. To find P (x), have a total of x dots composing the target shape. P (x) then equals the total number of dots in the shape. A 0-dimensional triangle is a point and a 1-dimensional triangle is simply a line, and therefore P (x) = 1 and P (x) = x, which is the sequence of natural numbers. The number of dots in each layer corresponds to P (x).