No, the name has no typo. 'tabl' is a keyword in the Online Encyclopedia of Integer Sequences (OEIS). It classifies a sequence as a regular triangular array of numbers.
The goal of this library is to identify semantically meaningful clusters around integer triangles, to implement these triangles, and to provide tools to study them.
The implementations are written in annotated Python and refer to a uniform interface.
An integer triangle is a list of integer lists (list[list[int]]). As usual in Python, all lists are 0-based; in particular, all triangles have offset = 0.
This allows us to identify an integer triangle with a list whose indexing is like this:
[ [(0,0)], [(1,0), (1,1)], [(2,0), (2,1), (2,2)], ..., [(n,0), (n,1), ... (n,k), ... (n,n)] ]
For example, G. W. Leibniz wrote down a number triangle in his "Dissertatio de arte combinatoria", Leipzig in 1666, that we will display like this:
[0] [0]
[1] [0, 1]
[2] [0, 2, 2]
[3] [0, 3, 4, 3]
[4] [0, 4, 6, 6, 4]
[5] [0, 5, 8, 9, 8, 5]
[6] [0, 6, 10, 12, 12, 10, 6]
[7] [0, 7, 12, 15, 16, 15, 12, 7]
[8] [0, 8, 14, 18, 20, 20, 18, 14, 8]
[9] [0, 9, 16, 21, 24, 25, 24, 21, 16, 9]
The first (left) column indicates the row number and is not part of the triangle.
To use the library, put the file Tables.py in the same directory where your project is or somewhere else where the interpreter can find it. Make sure your Python has the "more_itertools" package installed. The other Python files are not needed as long as you do not want to make your own additions to the library.
Test the installation.
from Tables import TablesListPreview
TablesListPreview()
This shows the list of the sequences implemented.
To use a Table from the library:
from Tables import Abel, InspectTable
InspectTable(Abel)
Use the Table as an iterable:
AbelRows = Abel.itr(7)
for row in AbelRows:
print(row, 'sum:', sum(row))
Go to the file "Template.py". It describes how you can add your own Table class to the library in three simple steps.
Currently, 116 triangles are included in this library implementing the class Table that provides the following methods:
__call__(self, n: int, k: int) -> int | T(n, k)
val (n:int, k:int) -> int | T(n, k)
row (n: int) -> tblrow | n-th row of table
itr (size: int) -> Iterator[list[int]] | traverse the first 'size' rows
flat (size: int) -> list[int] | flattened form of the first size rows
diag(n, size: int) -> list[int] | diagonal starting at the left side
col (k, size: int) -> list[int] | k-th column starting at the main diagonal
sum (size: int) -> list[int] | sums of the first size rows
antidiag (size: int) -> list[int] | upward anti-diagonals
rev (size: int) -> tblrow | reversed rows
acc (size: int) -> tblrow | accumulated row
alt (size: int) -> tblrow | alternating signs
diff (size: int) -> tblrow | first difference of row
der (size: int) -> tblrow | derivative of row
tab (size: int) -> tabl | table with size rows
mat (size: int) -> tabl | matrix form of lower triangular array
inv (size: int) -> tabl | inverse table
revinv (size: int) -> tabl | row reversed inverse
invrev (size: int) -> tabl | inverse of row reversed
off (N: int, K: int) -> rowgen | new offset (N, K)
revinv11 (size: int) -> tabl | revinv from offset (1, 1)
invrev11 (size: int) -> tabl | invrev from offset (1, 1)
poly(n: int, x: int) -> int | sum(T(n, k) * x^j for j=0..n)
trans(s: seq, size) -> list[int] | linear transformation induced by T
invtrans(s: seq, size) -> list[int] | inverse transformation induced by T
show (size: int) -> None | prints the first 'size' rows with row-numbers
These methods provide various functionalities for manipulating and generating integer triangles. For example, the row method returns the n-th row of the triangle, the rev method returns the reversed row of the triangle, and the antidiag method returns the n-th antidiagonal of the triangle.
This project aimed to identify the crucial triangles without relying on subjective assessment. For this, a metric had to be developed. We based the ranking on the number of distinct sequences generated by a fixed pool of elementary transformations mapping the triangles to other sequences. These transformations are called 'table traits' and are implemented in the module '_tabltraits'. These are very simple transformations that occur everywhere in the OEIS, but are not always recognized as such and then often lead to unnecessarily complicated formulas.
In the table below, the two columns on the right are examples of hundreds of similar identities that can be generated with the module. Note: References to OEIS-Ids are approximate. They often differ in signs and offset, sometimes also in the first few values.
| Name | Formula | Triangle | Trait |
|---|---|---|---|
| Triangle | $ T_{n,k} $ | Abel | A137452 |
| Tinv | $ (T^{-1})_{n,k} $ | Abel | A059297 |
| Trev | $ T_{n,n-k} $ | StirlingSet | A106800 |
| Tinvrev | $ (T_{n,n-k})^{-1} $ | FallingFactorial | A132013 |
| Trevinv | $ (T^{-1})_{n,n-k} $ | DyckPaths | A054142 |
| Toff11 | $ T_{n+1,k+1} $ | StirlingSet | A008277 |
| Trev11 | $ T_{n+1,n-k+1} $ | Eulerian | A008292 |
| Tinv11 | $ (T^{-1})_{n+1,k+1} $ | Eulerian | A055325 |
| Tinvrev11 | $ (T_{n+1,n-k+1})^{-1} $ | Eulerian | A055325 |
| Trevinv11 | $ (T^{-1})_{n+1,n-k+1} $ | StirlingSet | A094638 |
| Tantidiag | $ T_{n-k,k}\ \ (k \le n/2) $ | Motzkin | A106489 |
| Tacc | $ \sum_{j=0}^{k} T_{n,j} $ | Binomial | A008949 |
| Talt | $ T_{n,k} (-1)^{k} $ | Binomial | A130595 |
| Tder | $ T_{n,k+1}\ (k+1) $ | Abel | A225465 |
| TablCol0 | $ T_{n ,0} $ | WardCycle | A000007 |
| TablCol1 | $ T_{n+1,1} $ | WardCycle | A000142 |
| TablCol2 | $ T_{n+2,2} $ | WardCycle | A000276 |
| TablCol3 | $ T_{n+3,3} $ | WardCycle | A000483 |
| TablDiag0 | $ T_{n ,n} $ | WardSet | A001147 |
| TablDiag1 | $ T_{n+1,n} $ | WardSet | A000457 |
| TablDiag2 | $ T_{n+2,n} $ | WardSet | A000497 |
| TablDiag3 | $ T_{n+3,n} $ | WardSet | A000504 |
| TablLcm | $ \text{lcm} { \ | T_{n,k} | : k=0..n } $ | Binomial | A002944 |
| TablGcd | $ \text{gcd} { \ | T_{n,k} | : k=0..n } $ | Fubini | A141056 |
| TablMax | $ \text{max} { \ | T_{n,k} | : k=0..n } $ | BinaryPell | A109388 |
| TablSum | $ \sum_{k=0}^{n} T_{n,k} $ | Binomial | A000079 |
| EvenSum | $ \sum_{k=0}^{n} T_{n,k}\ (2 \mid k) $ | Binomial | A011782 |
| OddSum | $ \sum_{k=0}^{n} T_{n,k}\ (1 - (2 \mid k)) $ | Binomial | A131577 |
| AltSum | $ \sum_{k=0}^{n} T_{n,k} (-1)^{k} $ | Binomial | A000007 |
| AbsSum | $ \sum_{k=0}^{n} | T_{n,k} | $ | EulerTan | A009739 |
| AccSum | $ \sum_{k=0}^{n} \sum_{j=0}^{k} T_{n,j} $ | Binomial | A001792 |
| AccRevSum | $ \sum_{k=0}^{n} \sum_{j=0}^{k} T_{n,n-j} $ | StirlingCycle | A000774 |
| AntiDSum | $ \sum_{k=0}^{n/2} T_{n-k, k} $ | Binomial | A000045 |
| ColMiddle | $ T_{n, n / 2} $ | Binomial | A001405 |
| CentralE | $ T_{2 n, n} $ | Binomial | A000984 |
| CentralO | $ T_{2 n + 1, n} $ | Binomial | A001700 |
| PosHalf | $ \sum_{k=0}^{n}T_{n,k}\ 2^{n-k} $ | FallingFactorial | A010842 |
| NegHalf | $ \sum_{k=0}^{n}T_{n,k}\ (-2)^{n-k} $ | FallingFactorial | A000023 |
| TransNat0 | $ \sum_{k=0}^{n}T_{n,k}\ k $ | Binomial | A001787 |
| TransNat1 | $ \sum_{k=0}^{n}T_{n,k}\ (k+1) $ | Binomial | A001792 |
| TransSqrs | $ \sum_{k=0}^{n}T_{n,k}\ k^{2} $ | Lah | A103194 |
| BinConv | $ \sum_{k=0}^{n}T_{n,k}\ \binom{n}{k} $ | FallingFactorial | A002720 |
| InvBinConv | $ \sum_{k=0}^{n}T_{n,k}\ (-1)^{n-k}\ \binom{n}{k} $ | FallingFactorial | A009940 |
| PolyRow1 | $ \sum_{k=0}^{1}T_{1,k}\ n^k $ | Lucas | A005408 |
| PolyRow2 | $ \sum_{k=0}^{2}T_{2,k}\ n^k $ | Lucas | A000384 |
| PolyRow3 | $ \sum_{k=0}^{3}T_{3,k}\ n^k $ | Lucas | A015237 |
| PolyCol2 | $ \sum_{k=0}^{n}T_{n,k}\ 2^k $ | Abel | A007334 |
| PolyCol3 | $ \sum_{k=0}^{n}T_{n,k}\ 3^k $ | Abel | A362354 |
| PolyDiag | $ \sum_{k=0}^{n}T_{n,k}\ n^k $ | Abel | A193678 |
In Visual Studio Code, the TeX formulas are displayed correctly in the preview.
The module provides a central function that extracts the traits from tables and simultaneously searches for further information in the OEIS database. In particular, the function returns the A-number of the sequence, provided it is in the database.
def LookUp(t: Table, tr: Trait, info: bool = True) -> int:
"""
Look up the A-number in the OEIS database based on a trait of a table.
Args:
t (Table): The table to be analyzed.
tr (Trait): A function that extracts a trait from the table.
info (bool, optional): If True, information about the matching will be displayed. Defaults to True.
Returns:
int: The A-number of the sequence if found, otherwise 0.
Raises:
Exception: If the OEIS server cannot be reached after multiple attempts.
Currently, the function will return -999999 if the OEIS server cannot be reached.
Example:
>>> LookUp(Fubini, PolyDiag)
If info = False then the function returns 94420.
Otherwise it additionally displays information about the matching:
You searched: 1,1,10,219,8676,...
OEIS-data is: 1,1,10,219,8676,...
Info: Starting at index 0 the next 13 consecutive terms match.
The matched substring starts at 0 and has length 135.
*** Found: A094420 Generalized ordered Bell numbers Bo(n,n).
"""
With currently 70 implemented traits and 100 triangles, right out of the box 7000 queries can be made in this way. This not only systematically links the triangles with sequences to a network, but often reveals previously unnoticed connections.
In this way, we found that each of the 50 most highly ranked triangles generates, on average, 35 distinct related sequences registered in the OEIS. Thus, a high rank means the triangle is a significant structural component in the OEIS database and binds seemingly unrelated sequences into a semantically meaningful cluster.
Currently, 116 triangles are included in this ranking. The top 50 triangles are listed below. (Note that some links are not yet implemented.)
| Name | OEIS | Distinct | Hits | Traits | Links | |
|---|---|---|---|---|---|---|
| 1 | StirlingSet | A048993 | 51 | 60 | T | L |
| 2 | FallingFactorial | A008279 | 48 | 59 | T | L |
| 3 | Lah | A271703 | 46 | 54 | T | L |
| 4 | BinaryPell | A038207 | 45 | 54 | T | L |
| 5 | Lucas | A029635 | 44 | 59 | T | L |
| 6 | Partition | A072233 | 43 | 53 | T | L |
| 7 | Fubini | A131689 | 42 | 49 | T | L |
| 8 | StirlingCycle | A132393 | 42 | 60 | T | L |
| 9 | CatalanInv | A128908 | 41 | 47 | T | L |
| 10 | Ordinals | A002262 | 40 | 65 | T | L |
| 11 | DyckPathsInv | A085478 | 40 | 52 | T | L |
| 12 | Powers | A004248 | 39 | 48 | T | L |
| 13 | Motzkin | A064189 | 38 | 47 | T | L |
| 14 | BesselInv | A122848 | 37 | 46 | T | L |
| 15 | DyckPaths | A039599 | 37 | 48 | T | L |
| 16 | Eulerian | A173018 | 37 | 55 | T | L |
| 17 | Catalan | A128899 | 36 | 42 | T | L |
| 18 | Laguerre | A021009 | 36 | 42 | T | L |
| 19 | Monotone | A059481 | 36 | 50 | T | L |
| 20 | Narayana | A090181 | 36 | 54 | T | L |
| 21 | AbelInv | A059297 | 35 | 40 | T | L |
| 22 | Divisibility | A113704 | 35 | 55 | T | L |
| 23 | Abel | A137452 | 34 | 39 | T | L |
| 24 | Nicomachus | A036561 | 34 | 41 | T | L |
| 25 | OrderedCycle | A225479 | 34 | 38 | T | L |
| 26 | Schroeder | A122538 | 34 | 42 | T | L |
| 27 | Bessel | A132062 | 33 | 39 | T | L |
| 28 | BinomialBell | A056857 | 33 | 39 | T | L |
| 29 | BinomialCatalan | A098474 | 33 | 38 | T | L |
| 30 | CatalanPaths | A053121 | 33 | 50 | T | L |
| 31 | ChebyshevS | A168561 | 33 | 47 | T | L |
| 32 | LeibnizTable | A094053 | 32 | 51 | T | L |
| 33 | Rencontres | A008290 | 32 | 42 | T | L |
| 34 | LucasInv | A112857 | 31 | 41 | T | L |
| 35 | Naturals | A000027 | 31 | 35 | T | L |
| 36 | Worpitzky | A028246 | 31 | 41 | T | L |
| 37 | SchroederP | A104684 | 30 | 34 | T | L |
| 38 | ChebyshevT | A053120 | 29 | 43 | T | L |
| 39 | MotzkinInv | A101950 | 28 | 39 | T | L |
| 40 | MotzkinPoly | A359364 | 28 | 41 | T | L |
| 41 | PartitionDist | A000000 | 28 | 41 | T | L |
| 42 | Bell | A011971 | 27 | 37 | T | L |
| 43 | ChebyshevU | A115322 | 27 | 38 | T | L |
| 44 | Euler | A109449 | 27 | 36 | T | L |
| 45 | StirlingCycleB | A028338 | 27 | 38 | T | L |
| 46 | Binomial | A007318 | 26 | 68 | T | L |
| 47 | DoublePochhammer | A039683 | 26 | 32 | T | L |
| 48 | Pascal | A007318 | 26 | 68 | T | L |
| 49 | Bessel2 | A359760 | 25 | 41 | T | L |
| 50 | Delannoy | A008288 | 25 | 49 | T | L |