# Binomial triangle

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Binomial triangles are a special case of ${\displaystyle p-}$nomial triangles, where ${\displaystyle p = 2}$; this means, that two starting points are needed to construct the triangle. The most basic binomial triangle is Pascal's triangle as it establishes the binomial distribution. There are many kinds of binomial triangles as they can be established via special numbers (in example primes or mathematical constants), sequences (in example the sequence of any combinatorial zeta function), patterns (in example alternating ${\displaystyle -}$ or recurring numbers) or any other mathematical construction regulation.

## Definition

Let ${\displaystyle n}$, ${\displaystyle k \in \mathbb{K}}$, whereby ${\displaystyle \mathbb{K}}$ is some arbitrary non empty set. Then the entry in the ${\displaystyle n-}$th row and ${\displaystyle k-}$th column of its binomial triangle is denoted by:

${\displaystyle \displaystyle\binom{-n}{-k}_{\mathbb{K}} }$

where:

${\displaystyle \begin{array}{lll} \displaystyle\binom{0}{0}_{\mathbb{K}} := 0\text{,} & \displaystyle\binom{-n}{0}_{\mathbb{K}} := q_{n} \in \mathbb{K}\text{,} & \displaystyle\binom{-n}{-n}_{\mathbb{K}} := w_{n} \in \mathbb{K} \end{array} }$

Further, the ${\displaystyle \mathbb{K}}$ triangle suffices either:

${\displaystyle \displaystyle\binom{n + 1}{k + 1}_{\mathbb{K}} = \displaystyle\binom{n}{k}_{\mathbb{K}} \pm \displaystyle\binom{n}{k + 1}_{\mathbb{K}}}$

or:

${\displaystyle \displaystyle\binom{-n - 1}{-k - 1}_{\mathbb{K}} = \displaystyle\binom{-n}{-k}_{\mathbb{K}} \pm \displaystyle\binom{-n}{-k - 1}_{\mathbb{K}}}$

for all ${\displaystyle k}$, ${\displaystyle n \geq 2}$, but not necessarily. With those binomial coefficients the triangles can be visualized as follows:

${\displaystyle \begin{array}{ccccccccc} \dots & & \dots & & \dots & & \dots & & \dots \\ & & & & & & & & \\ & \displaystyle\binom{\pm 3}{0}_{s_{n}} & & \displaystyle\binom{\pm 3}{\pm 1}_{s_{n}} & & \displaystyle\binom{\pm 3}{\pm 2}_{s_{n}} & & \displaystyle\binom{\pm 3}{\pm 3}_{s_{n}} &\\ & & & & & & & & \\ & & \displaystyle\binom{\pm 2}{0}_{s_{n}} & & \displaystyle\binom{\pm 2}{\pm 1}_{s_{n}} & & \displaystyle\binom{\pm 2}{\pm 2}_{s_{n}} & & \\ & & & & & & & & \\ & & & \displaystyle\binom{\pm 1}{0}_{s_{n}} & & \displaystyle\binom{\pm 1}{\pm 1}_{s_{n}} & & & \\ & & & & & & & \\ & & & & \hspace{4pt}\displaystyle\binom{0}{0}_{s_{n}} & & & & \\ \end{array}}$

which continues with ${\displaystyle n = 4}$, ${\displaystyle 5}$, ${\displaystyle \dots}$; the values equivalent to the binomial coefficients depend on the elements ${\displaystyle q_{n}}$, ${\displaystyle w_{n} \in \mathbb{K}}$.

## Remark: value for zeroth binomial coefficient

The value for:

${\displaystyle \displaystyle\binom{0}{0}_{\mathbb{K}} }$

is arbitrary chosen. Therefore, it can be any complex value, although it is convention to choose the entry to be equal to one or zero.

## Conclusion: all possible kind of binomial triangles

From the definition follows immediately, that binomial triangles can be established from elements of any kind of arbitrary set of numbers. It is only necessary to equate that set with ${\displaystyle \mathbb{K}}$, even the empty set, which results in a zero binomial triangle. This is another confirmation for the theorem that secures the existence of the unit (${\displaystyle -}$ and the zero) ${\displaystyle p-}$nomial triangles for all ${\displaystyle p}$.

## Remark: what binomial triangles are not

The binomial triangles have not to be confused with the expansion of the binomial coefficients to real ${\displaystyle -}$ and complex numbers. That is something different entirely.

## ${\displaystyle \mathbb{N}}$ binomial triangles

The natural (number) binomial triangles are established via starting values that are solely natural numbers ${\displaystyle \mathbb{N}}$. The most important ${\displaystyle \mathbb{N}}$ binomial triangle is the natural number binomial triangle as the underlying sequence is the inverse harmonic series.

## ${\displaystyle \mathbb{P}}$ binomial triangles

The prime binomial triangles are established via starting values that are exclusively prime numbers. This does not mean, that only prime numbers appear in the rows. The most important ${\displaystyle \mathbb{P}}$ binomial triangle is the (positive) prime number binomial triangle as the underlying sequence is the inverse prime zeta function.

## ${\displaystyle \mathbb{C}}$ binomial triangles

Complex binomial triangles are established via starting values that are complex numbers. The most basic complex binomial triangle is the complex Pascal triangle as it establishes the complex binomial distribution.

## Alternating binomial triangles

Alternating binomial triangles are established via starting values that alternate between postive and negative.

## ${\displaystyle p-q-}$adic binomial triangles

${\displaystyle p-q-}$adic binomial triangles are binomial triangles whose entries are represented in a ${\displaystyle p-q-}$adic number base.

## ${\displaystyle s_{n}}$ binomial triangles

Sequence binomial triangles are established via a sequence, which means, that they can be a second, underlying feature for establishing a binomial triangle. One such example is the (positive) ${\displaystyle s_{n}^{\kappa} \in \zeta(z)}$ ${\displaystyle -}$, with ${\displaystyle \operatorname{Re}(s_{n}^{\kappa}) > 1}$, binomial triangle, where the underlying sequence is the Riemann zeta function; another such example is ${\displaystyle p_{n}^{\kappa} \in p(z)}$ ${\displaystyle -}$, with ${\displaystyle \operatorname{Re}(p_{n}^{\kappa}) > 1}$, binomial triangle where the underlying sequence is the prime zeta function.