Editing Pascal's triangle (section)

=== Diagonals ===
[[File:Pascal_triangle_simplex_numbers.svg|thumb|upright=1.25|Derivation of [[simplex]] numbers from a left-justified Pascal's triangle]]
The diagonals of Pascal's triangle contain the [[Figurate numbers#Triangular numbers and their analogs in higher dimensions|figurate numbers]] of simplices:
* The diagonals going along the left and right edges contain only 1's.
* The diagonals next to the edge diagonals contain the [[natural number]]s in order. The 1-dimensional simplex numbers increment by 1 as the line segments extend to the next whole number along the number line.
* Moving inwards, the next pair of diagonals contain the [[triangular number]]s in order.
* The next pair of diagonals contain the [[tetrahedral number]]s in order, and the next pair give [[pentatope number]]s.

::<math>\begin{align}
P_0(n) &= P_d(0) = 1, \\
P_d(n) &= P_d(n-1) + P_{d-1}(n) \\
&= \sum_{i=0}^n P_{d-1}(i) = \sum_{i=0}^d P_i(n-1).
\end{align}</math>

The symmetry of the triangle implies that the ''n''<sup>th</sup> d-dimensional number is equal to the ''d''<sup>th</sup> ''n''-dimensional number.

An alternative formula that does not involve recursion is
<math display="block">P_d(n)=\frac{1}{d!}\prod_{k=0}^{d-1} (n+k) = {n^{(d)}\over d!} = \binom{n+d-1}{d},</math>
where ''n''<sup>(''d'')</sup> is the [[rising factorial]].

The geometric meaning of a function ''P''<sub>''d''</sub> is: ''P''<sub>''d''</sub>(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''<sub>''d''</sub>(1) = 1. Place these dots in a manner analogous to the placement of numbers in Pascal's triangle. To find P<sub>''d''</sub>(''x''), have a total of ''x'' dots composing the target shape. P<sub>''d''</sub>(''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''<sub>0</sub>(''x'') = 1 and ''P''<sub>1</sub>(''x'') = ''x'', which is the sequence of natural numbers. The number of dots in each layer corresponds to ''P''<sub>''d''&nbsp;−&nbsp;1</sub>(''x'').