Id

Symbol

Conducement

Reducementals

Inverse

Uninomial

Priming:

$\langle 1\rangle \circ \left\{1\right\}=\left\{1\right\}$

-

-

1

${\mathbb{U}}_0$

$\left\{1\right\}\oplus \left\{\begin{array}{ccc}1& 1& \begin{array}{cc}1& \cdots \end{array}\end{array}\right\}=\left\{\begin{array}{ccc}1& 1& \begin{array}{cc}1& \cdots \end{array}\end{array}\right\}$

$\left\{1\right\}\odot \left\{\left(\begin{array}{c}n\\ n\end{array}\right)\right\}$

𝕦₀

$\left\{1\right\}$

2

${\mathbb{U}}_1$

$\begin{array}{l}\left\{\begin{array}{ccc}1& 1& \begin{array}{cc}1& \cdots \end{array}\end{array}\right\}\oplus \left\{\begin{array}{ccc}1& 1& \begin{array}{cc}1& \cdots \end{array}\end{array}\right\}\\ =\left\{\begin{array}{ccc}1& 2& \begin{array}{cc}3& \begin{array}{cc}\cdots & \left(\begin{array}{c}n\\ n-1\end{array}\right)\end{array}\end{array}\end{array}\right\}\end{array}$

$\left\{\begin{array}{cc}1& 1\end{array}\right\}\odot \left\{\begin{array}{cc}\left(\begin{array}{c}n\\ n\end{array}\right)& \left(\begin{array}{c}n\\ n-1\end{array}\right)\end{array}\right\}$

𝕦₋₁

$\left\{\begin{array}{cc}1& -1\end{array}\right\}$

3

${\mathbb{U}}_2$

$\begin{array}{l}\left\{\begin{array}{ccc}1& 2& \begin{array}{cc}3& \begin{array}{cc}\cdots & \left(\begin{array}{c}n\\ n-1\end{array}\right)\end{array}\end{array}\end{array}\right\}\oplus \left\{\begin{array}{ccc}1& 1& \begin{array}{cc}1& \cdots \end{array}\end{array}\right\}\\ =\left\{\begin{array}{ccc}1& 3& \begin{array}{cc}6& \begin{array}{cc}\cdots & \left(\begin{array}{c}n+1\\ n-1\end{array}\right)\end{array}\end{array}\end{array}\right\}\end{array}$

$\left\{\begin{array}{cc}1& \begin{array}{cc}2& 1\end{array}\end{array}\right\}\odot \left\{\begin{array}{cc}\left(\begin{array}{c}n\\ n\end{array}\right)& \begin{array}{cc}\left(\begin{array}{c}n\\ n-1\end{array}\right)& \left(\begin{array}{c}n\\ n-2\end{array}\right)\end{array}\end{array}\right\}$

𝕦₋₂

$\left\{\begin{array}{cc}1& \begin{array}{cc}-2& 1\end{array}\end{array}\right\}$

4

${\mathbb{U}}_3$

$\begin{array}{l}\left\{\begin{array}{ccc}1& 3& \begin{array}{cc}6& \begin{array}{cc}\cdots & \left(\begin{array}{c}n+1\\ n-1\end{array}\right)\end{array}\end{array}\end{array}\right\}\oplus \left\{\begin{array}{ccc}1& 1& \begin{array}{cc}1& \cdots \end{array}\end{array}\right\}\\ =\left\{\begin{array}{ccc}1& 4& \begin{array}{cc}10& \begin{array}{cc}\cdots & \left(\begin{array}{c}n+2\\ n-1\end{array}\right)\end{array}\end{array}\end{array}\right\}\end{array}$

$\left\{\begin{array}{cc}1& \begin{array}{cc}3& \begin{array}{cc}3& 1\end{array}\end{array}\end{array}\right\}\odot \left\{\begin{array}{cc}\left(\begin{array}{c}n\\ n\end{array}\right)& \begin{array}{cc}\left(\begin{array}{c}n\\ n-1\end{array}\right)& \begin{array}{cc}\left(\begin{array}{c}n\\ n-2\end{array}\right)& \left(\begin{array}{c}n\\ n-3\end{array}\right)\end{array}\end{array}\end{array}\right\}$

𝕦₋₃

$\left\{\begin{array}{cc}1& \begin{array}{cc}-3& \begin{array}{cc}3& -1\end{array}\end{array}\end{array}\right\}$

Binomial

5

Priming:

$\langle \begin{array}{cc}1& 1\end{array}\rangle \circ \left\{\begin{array}{cc}1& 1\end{array}\right\}=\left\{\begin{array}{cc}1& 1\end{array}\right\}$

6^*

${\mathbb{B}}_1$

$\begin{array}{l}\left\{\begin{array}{cc}1& 1\end{array}\right\}\oplus \left\{\begin{array}{ccc}1& 2& \begin{array}{cc}3& \begin{array}{cc}\cdots & \left(\begin{array}{c}n\\ n-1\end{array}\right)\end{array}\end{array}\end{array}\right\}\\ =\left\{\begin{array}{ccc}1& 3& \begin{array}{ccc}5& 7& \cdots \end{array}\end{array}\right\}\end{array}$

$\left\{\begin{array}{cc}1& 1\end{array}\right\}\odot \left\{\begin{array}{cc}\left(\begin{array}{c}n-1\\ n-1\end{array}\right)& \left(\begin{array}{c}n-1\\ n-2\end{array}\right)\end{array}\right\}$

𝕓₋₁

$\left\{\begin{array}{cc}1& \begin{array}{cc}0& -1\end{array}\end{array}\right\}$

7

${\mathbb{B}}_2$

$\begin{array}{l}\left\{\begin{array}{ccc}1& 3& \begin{array}{ccc}5& 7& \cdots \end{array}\end{array}\right\}\oplus \left\{\begin{array}{cc}1& 1\end{array}\right\}\\ =\left\{\begin{array}{ccc}1& 4& \begin{array}{ccc}8& 12& \cdots \end{array}\end{array}\right\}\end{array}$

$\left\{\begin{array}{cc}1& \begin{array}{cc}2& 1\end{array}\end{array}\right\}\odot \left\{\begin{array}{cc}\left(\begin{array}{c}n-1\\ n-1\end{array}\right)& \begin{array}{cc}\left(\begin{array}{c}n-1\\ n-2\end{array}\right)& \left(\begin{array}{c}n-1\\ n-3\end{array}\right)\end{array}\end{array}\right\}$

𝕓₋₂

$\left\{\begin{array}{cc}1& \begin{array}{cc}1& \begin{array}{cc}-1& -1\end{array}\end{array}\end{array}\right\}$

8

${\mathbb{B}}_3$

$\begin{array}{l}\left\{\begin{array}{ccc}1& 4& \begin{array}{ccc}8& 12& \cdots \end{array}\end{array}\right\}\oplus \left\{\begin{array}{cc}1& 1\end{array}\right\}\\ =\left\{\begin{array}{ccc}1& 5& \begin{array}{ccc}12& 20& \cdots \end{array}\end{array}\right\}\end{array}$

$\left\{\begin{array}{cc}1& \begin{array}{cc}3& \begin{array}{cc}3& 1\end{array}\end{array}\end{array}\right\}\odot \left\{\begin{array}{cc}\left(\begin{array}{c}n-1\\ n-1\end{array}\right)& \begin{array}{cc}\left(\begin{array}{c}n-1\\ n-2\end{array}\right)& \begin{array}{cc}\left(\begin{array}{c}n-1\\ n-3\end{array}\right)& \left(\begin{array}{c}n-1\\ n-4\end{array}\right)\end{array}\end{array}\end{array}\right\}$

𝕓₋₃

$\left\{\begin{array}{cc}1& \begin{array}{cc}2& \begin{array}{cc}0& \begin{array}{cc}-2& -1\end{array}\end{array}\end{array}\end{array}\right\}$

Trinomial Sequences

9

${\mathbb{T}}_1$

$\begin{array}{l}\left\{\begin{array}{cc}1& \begin{array}{cc}1& 1\end{array}\end{array}\right\}\oplus \left\{\begin{array}{ccc}1& 2& \begin{array}{ccc}3& 4& \cdots \end{array}\end{array}\right\}\\ =\left\{\begin{array}{ccc}1& 3& \begin{array}{ccc}6& 9& \cdots \end{array}\end{array}\right\}\end{array}$

$\left\{\begin{array}{cc}1& \begin{array}{cc}1& 1\end{array}\end{array}\right\}\odot \left\{\begin{array}{cc}\left(\begin{array}{c}n-2\\ n-2\end{array}\right)& \begin{array}{cc}\left(\begin{array}{c}n-2\\ n-3\end{array}\right)& \left(\begin{array}{c}n-2\\ n-4\end{array}\right)\end{array}\end{array}\right\}$

𝕥₋₁

$\left\{\begin{array}{cc}1& \begin{array}{cc}0& \begin{array}{cc}0& -1\end{array}\end{array}\end{array}\right\}$

10

${\mathbb{T}}_2$

$\begin{array}{l}\left\{\begin{array}{cc}1& \begin{array}{cc}1& 1\end{array}\end{array}\right\}\oplus \left\{\begin{array}{ccc}1& 3& \begin{array}{ccc}6& 9& \cdots \end{array}\end{array}\right\}\\ =\left\{\begin{array}{ccc}1& 4& \begin{array}{ccc}10& 18& \cdots \end{array}\end{array}\right\}\end{array}$

$\begin{array}{l}\left\{\begin{array}{cc}1& \begin{array}{cc}2& \begin{array}{cc}3& \begin{array}{cc}2& 1\end{array}\end{array}\end{array}\end{array}\right\}\\ \odot \left\{\begin{array}{cc}\left(\begin{array}{c}n-2\\ n-2\end{array}\right)& \begin{array}{cc}\left(\begin{array}{c}n-2\\ n-3\end{array}\right)& \begin{array}{cc}\left(\begin{array}{c}n-2\\ n-4\end{array}\right)& \begin{array}{cc}\left(\begin{array}{c}n-2\\ n-5\end{array}\right)& \left(\begin{array}{c}n-2\\ n-6\end{array}\right)\end{array}\end{array}\end{array}\end{array}\right\}\end{array}$

𝕥₋₂

$\left\{\begin{array}{cc}1& \begin{array}{cc}1& \begin{array}{cc}1& \begin{array}{cc}-1& \begin{array}{cc}-1& -1\end{array}\end{array}\end{array}\end{array}\end{array}\right\}$

11

${\mathbb{T}}_3$

$\begin{array}{l}\left\{\begin{array}{cc}1& \begin{array}{cc}1& 1\end{array}\end{array}\right\}\oplus \left\{\begin{array}{ccc}1& 4& \begin{array}{ccc}10& 18& \cdots \end{array}\end{array}\right\}\\ =\left\{\begin{array}{ccc}1& 5& \begin{array}{ccc}15& 32& \cdots \end{array}\end{array}\right\}\end{array}$

$\begin{array}{l}\left\{\begin{array}{cc}1& \begin{array}{cc}3& \begin{array}{cc}6& \begin{array}{cc}7& \begin{array}{cc}6& \begin{array}{cc}3& 1\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\right\}\\ \odot \{\begin{array}{cc}\left(\begin{array}{c}n-2\\ n-2\end{array}\right)& \begin{array}{cc}\left(\begin{array}{c}n-2\\ n-3\end{array}\right)& \begin{array}{cc}\left(\begin{array}{c}n-2\\ n-4\end{array}\right)& \begin{array}{c}\left(\begin{array}{c}n-2\\ n-5\end{array}\right)\end{array}\end{array}\end{array}\end{array}\\ \begin{array}{cc}\left(\begin{array}{c}n-2\\ n-6\end{array}\right)& \begin{array}{cc}\left(\begin{array}{c}n-2\\ n-7\end{array}\right)& \left(\begin{array}{c}n-2\\ n-8\end{array}\right)\end{array}\end{array}\}\end{array}$

𝕥₋₃

$\left\{\begin{array}{cc}1& \begin{array}{cc}2& \begin{array}{cc}3& \begin{array}{cc}1& \begin{array}{cc}-1& \begin{array}{cc}-3& \begin{array}{cc}-2& -1\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\right\}$