Pascal's triangle
E26830
Pascal's triangle is a triangular array of numbers in which each entry is the sum of the two directly above it, widely used in combinatorics, algebra, and probability.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Pascal's triangle canonical | 5 |
| Khayyam triangle | 1 |
| Traité du triangle arithmétique | 1 |
| Yang Hui's triangle | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T209661 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Pascal's triangle Context triple: [binomial theorem, relatesTo, Pascal's triangle]
-
A.
binomial theorem
The binomial theorem is a fundamental algebraic formula that provides a systematic way to expand powers of binomial expressions, playing a key role in combinatorics and mathematical analysis.
-
B.
Aitken
Aitken is a Scottish-origin surname notably borne by Max Aitken, 1st Baron Beaverbrook, a prominent Canadian-British newspaper magnate and politician.
-
C.
Mathematical Bridge
The Mathematical Bridge is a famous wooden footbridge at Queens' College, Cambridge, known for its elegant arch that is constructed entirely from straight timbers.
-
D.
Jakob Bernoulli
Jakob Bernoulli was a pioneering Swiss mathematician of the late 17th century, renowned for his foundational work in calculus and probability theory, including the early formulation of the law of large numbers.
-
E.
Numbers
Numbers is the fourth book of the Hebrew Bible and the Christian Old Testament, recounting the Israelites’ wilderness wanderings and organizing laws and censuses.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Pascal's triangle Target entity description: Pascal's triangle is a triangular array of numbers in which each entry is the sum of the two directly above it, widely used in combinatorics, algebra, and probability.
-
A.
binomial theorem
The binomial theorem is a fundamental algebraic formula that provides a systematic way to expand powers of binomial expressions, playing a key role in combinatorics and mathematical analysis.
-
B.
Aitken
Aitken is a Scottish-origin surname notably borne by Max Aitken, 1st Baron Beaverbrook, a prominent Canadian-British newspaper magnate and politician.
-
C.
Mathematical Bridge
The Mathematical Bridge is a famous wooden footbridge at Queens' College, Cambridge, known for its elegant arch that is constructed entirely from straight timbers.
-
D.
Jakob Bernoulli
Jakob Bernoulli was a pioneering Swiss mathematician of the late 17th century, renowned for his foundational work in calculus and probability theory, including the early formulation of the law of large numbers.
-
E.
Numbers
Numbers is the fourth book of the Hebrew Bible and the Christian Old Testament, recounting the Israelites’ wilderness wanderings and organizing laws and censuses.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical object
ⓘ
number triangle ⓘ |
| boundaryCondition |
C(n,0) = 1 for all n ≥ 0
ⓘ
C(n,n) = 1 for all n ≥ 0 ⓘ |
| combinatorialMeaning |
C(n,k) counts k-element subsets of an n-element set
ⓘ
C(n,k) counts number of paths in a grid from (0,0) to (k,n-k) using unit steps ⓘ |
| constructionMethod | start with 1 at top and repeatedly add adjacent pairs to form next row ⓘ |
| containsOnly | nonnegative integers ⓘ |
| definingProperty | each entry is the sum of the two entries directly above it ⓘ |
| diagonalProperty |
first diagonal consists of all 1s
ⓘ
fourth diagonal lists tetrahedral numbers ⓘ second diagonal lists natural numbers n ⓘ third diagonal lists triangular numbers ⓘ |
| entryFormula | C(n,k) = n choose k ⓘ |
| entryNotation |
(n k)
ⓘ
C(n,k) ⓘ |
| field |
algebra
ⓘ
combinatorics ⓘ probability theory ⓘ |
| generalization |
multinomial theorem
ⓘ
surface form:
Pascal's pyramid
multinomial coefficients ⓘ |
| growthDirection | extends infinitely downward ⓘ |
| hasRow |
row 0: 1
ⓘ
row 1: 1 1 ⓘ row 2: 1 2 1 ⓘ row 3: 1 3 3 1 ⓘ row 4: 1 4 6 4 1 ⓘ |
| hasShape | triangular array ⓘ |
| historicalUseBeforePascal |
China
ⓘ
India ⓘ Persia ⓘ |
| identity |
sum_{k} C(n,k)^2 = C(2n,n)
ⓘ
∑_{k} (-1)^k C(n,k) = 0 for n ≥ 1 ⓘ |
| knownAsInChina |
Pascal's triangle
self-linksurface differs
ⓘ
surface form:
Yang Hui's triangle
|
| knownAsInPersia |
Pascal's triangle
ⓘ
surface form:
Khayyam triangle
|
| namedAfter | Blaise Pascal ⓘ |
| nthRowRepresents | coefficients of the binomial expansion of (x + y)^n ⓘ |
| parityPattern | mod 2 pattern forms Sierpiński triangle ⓘ |
| relatedTo |
Bernoulli trials
ⓘ
binomial coefficients ⓘ binomial theorem ⓘ |
| rowIndexingStartsAt | 0 ⓘ |
| rowSumProperty | sum of entries in nth row equals 2^n ⓘ |
| satisfiesRecurrence | C(n,k) = C(n-1,k-1) + C(n-1,k) ⓘ |
| symmetryProperty | C(n,k) = C(n,n-k) ⓘ |
| topElement | 1 ⓘ |
| usedFor |
computing binomial probabilities
ⓘ
counting combinations ⓘ expanding binomials ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Pascal's triangle Description of subject: Pascal's triangle is a triangular array of numbers in which each entry is the sum of the two directly above it, widely used in combinatorics, algebra, and probability.
Referenced by (8)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Yang Hui's triangle
this entity surface form:
Khayyam triangle
this entity surface form:
Traité du triangle arithmétique