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Infinite matrices with Pascal's triangle as elements
In matrix theory and combinatorics, a Pascal matrix is a matrix (possibly infinite) containing the binomial coefficients as its elements. It is thus an
Pascal_matrix
Triangular array of the binomial coefficients
entries in Pascal's triangle (Singmaster's conjecture) Pascal matrix Pascal's pyramid Pascal's simplex Proton NMR, one application of Pascal's triangle
Pascal's_triangle
Algebraic expansion of powers of a binomial
value). These coefficients for varying n and k can be arranged to form Pascal's triangle. These numbers also occur in combinatorics, where (nk) gives
Binomial_theorem
Product of consecutive factorial numbers
arrays Lozanić's triangle · Sierpinski triangle · Pascal matrix · Multiplicities of entries in Pascal's triangle Operators & calculus links Difference operator
Superfactorial
Argument for the belief in God
Pascal's wager is a philosophical argument advanced by Blaise Pascal (1623–1662), a French mathematician, philosopher, physicist, and theologian. This
Pascal's_wager
Probability distribution
theory and statistics, the negative binomial distribution, also called a Pascal distribution, is a discrete probability distribution that models the number
Negative binomial distribution
Negative_binomial_distribution
Type of permutation of a set of elements
arrays Lozanić's triangle · Sierpinski triangle · Pascal matrix · Multiplicities of entries in Pascal's triangle Operators & calculus links Difference operator
Derangement
Graphical aid for deriving some concepts in combinatorics
numbers for n = 0, 1, 2, 3, ... are those in the (k − 1)st diagonal of Pascal's triangle. For example, when k = 3 the nth number is the (n + 1)st triangular
Stars and bars (combinatorics)
Stars_and_bars_(combinatorics)
Number theory expression
arrays Lozanić's triangle · Sierpinski triangle · Pascal matrix · Multiplicities of entries in Pascal's triangle Operators & calculus links Difference operator
Legendre's_formula
Arrangement of trinomial coefficients
mathematics, Pascal's pyramid is a three-dimensional arrangement of the coefficients of the trinomial expansion and the trinomial distribution. Pascal's pyramid
Pascal's_pyramid
Number computed as a product of powers
arrays Lozanić's triangle · Sierpinski triangle · Pascal matrix · Multiplicities of entries in Pascal's triangle Operators & calculus links Difference operator
Hyperfactorial
Recursive mathematical formula
arrays Lozanić's triangle · Sierpinski triangle · Pascal matrix · Multiplicities of entries in Pascal's triangle Operators & calculus links Difference operator
Exponential_factorial
Probability distribution
Blaise Pascal had earlier considered the case where p = 1/2, tabulating the corresponding binomial coefficients in what is now recognized as Pascal's triangle
Binomial_distribution
Recurrence relations of binomial coefficients in Pascal's triangle
} The name stems from the graphical representation of the identity on Pascal's triangle: when the addends represented in the summation and the sum itself
Hockey-stick_identity
Mathematical theorem on convolved binomial coefficients
draws without replacement from an urn containing n red and m blue marbles. Pascal's identity Hockey-stick identity Rothe–Hagen identity See Askey, Richard
Vandermonde's_identity
Branch of discrete mathematics
mathematics and the sciences, combinatorics enjoyed a rebirth. Works of Pascal, Newton, Jacob Bernoulli and Leonhard Euler became foundational in the emerging
Combinatorics
Array of numbers
[19−13205−6] denotes a matrix with two rows and three columns. This is often referred to as a "two-by-three matrix", a 2 × 3 matrix, or a matrix of dimension 2 × 3
Matrix_(mathematics)
Recursive integer sequence
number of ways of associating n applications of a binary operator (as in the matrix chain multiplication problem). For n = 3, for example, we have the following
Catalan_number
matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called entries
List_of_named_matrices
Sequence of numbers ((2n) choose (n))
since they show up exactly in the middle of the even-numbered rows in Pascal's triangle. The first few central binomial coefficients starting at n = 0
Central_binomial_coefficient
Generalization of the binomial theorem to other polynomials
One can use the multinomial theorem to generalize Pascal's triangle or Pascal's pyramid to Pascal's simplex. This provides a quick way to generate a lookup
Multinomial_theorem
Formula in mathematics
can be defined with a generalization of Pascal's triangle to three dimensions, called Pascal's pyramid or Pascal's tetrahedron. The trinomial expansion can
Trinomial_expansion
Theorem on prime numbers
arrays Lozanić's triangle · Sierpinski triangle · Pascal matrix · Multiplicities of entries in Pascal's triangle Operators & calculus links Difference operator
Wilson's_theorem
Matrix equal to its conjugate-transpose
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex-valued entries that is equal to its own conjugate transpose
Hermitian_matrix
Conjecture in combinatorial number theory
mathematics Is there some constant N such that every entry (apart from 1) of Pascal's triangle appears fewer than N times? More unsolved problems in mathematics
Singmaster's_conjecture
Pascal's triangle Multiset Multivariate gamma function Narayana numbers Negative binomial distribution Nörlund–Rice integral Pascal matrix Pascal's pyramid
List of factorial and binomial topics
List_of_factorial_and_binomial_topics
Numerical method for calculating the gamma function
{1}{2}}\right)}e^{l+g+{\frac {1}{2}}}} where Cn,m represents the (n, m)th element of the matrix of coefficients for the Chebyshev polynomials, which can be calculated recursively
Lanczos_approximation
Mathematical functions
{\begin{aligned}\left[{\begin{matrix}n\\k\end{matrix}}\right]_{f,t}&=\left[x^{k-1}\right](x)_{n,f,t}\\&=f(n-1)t^{1-n}\left[{\begin{matrix}n-1\\k\end{matrix}}\right]_{f
Falling_and_rising_factorials
Numbering of combinations of items
"combinatorial number system" by Knuth (2011). He also references Ernesto Pascal (1887). The term "combinadic" is introduced by James McCaffrey. Unlike the
Combinatorial_number_system
Family of polynomials
∞ {\displaystyle m\to \infty } , both equations remain valid. The first Pascal analog allows computation of the Gaussian binomial coefficients recursively
Gaussian_binomial_coefficient
Generalization of the mathematical factorial
arrays Lozanić's triangle · Sierpinski triangle · Pascal matrix · Multiplicities of entries in Pascal's triangle Operators & calculus links Difference operator
Bhargava_factorial
matrices have an LDU decomposition involving triangular Pascal matrices and a diagonal matrix of the powers of 2. The eigenvalues are ± 2 n {\displaystyle
Krawtchouk_matrices
Mathematical result on arithmetic properties of binomial coefficients
coefficients forming each of the two triangles in the Star of David shape in Pascal's triangle are equal: gcd { ( n − 1 k − 1 ) , ( n k + 1 ) , ( n + 1 k ) }
Star_of_David_theorem
Generalization of the binomial distribution
covariance matrix of rank k − 1. In the special case where k = n and where the pi are all equal, the covariance matrix is the centering matrix. The entries
Multinomial_distribution
Uniqueness theorem in complex analysis
arrays Lozanić's triangle · Sierpinski triangle · Pascal matrix · Multiplicities of entries in Pascal's triangle Operators & calculus links Difference operator
Carlson's_theorem
Upper bound on intersecting set families
arrays Lozanić's triangle · Sierpinski triangle · Pascal matrix · Multiplicities of entries in Pascal's triangle Operators & calculus links Difference operator
Erdős–Ko–Rado_theorem
Mathematical function
arrays Lozanić's triangle · Sierpinski triangle · Pascal matrix · Multiplicities of entries in Pascal's triangle Operators & calculus links Difference operator
Digamma_function
Number of subsets of a given size
the left and right of Pascal's triangle, the entries (shown as blanks) are all zero. Pascal's rule also gives rise to Pascal's triangle: Row number n
Binomial_coefficient
Generalized chain rule in calculus
arrays Lozanić's triangle · Sierpinski triangle · Pascal matrix · Multiplicities of entries in Pascal's triangle Operators & calculus links Difference operator
Faà_di_Bruno's_formula
Mathematical series
arrays Lozanić's triangle · Sierpinski triangle · Pascal matrix · Multiplicities of entries in Pascal's triangle Operators & calculus links Difference operator
Binomial_series
arrays Lozanić's triangle · Sierpinski triangle · Pascal matrix · Multiplicities of entries in Pascal's triangle Operators & calculus links Difference operator
Binomial_QMF
Mathematical sequences in combinatorics
understood to be matrix inverse relationships. That is, let s be the lower triangular matrix of Stirling numbers of the first kind, whose matrix elements s
Stirling_number
Solved prime-number problem
arrays Lozanić's triangle · Sierpinski triangle · Pascal matrix · Multiplicities of entries in Pascal's triangle Operators & calculus links Difference operator
Proof_of_Bertrand's_postulate
Numeral system in combinatorics
arrays Lozanić's triangle · Sierpinski triangle · Pascal matrix · Multiplicities of entries in Pascal's triangle Operators & calculus links Difference operator
Factorial_number_system
Product of the first "n" prime numbers
arrays Lozanić's triangle · Sierpinski triangle · Pascal matrix · Multiplicities of entries in Pascal's triangle Operators & calculus links Difference operator
Primorial
Expectation or average of the falling factorial of a random variable
arrays Lozanić's triangle · Sierpinski triangle · Pascal matrix · Multiplicities of entries in Pascal's triangle Operators & calculus links Difference operator
Factorial_moment
Type of polynomial sequence
arrays Lozanić's triangle · Sierpinski triangle · Pascal matrix · Multiplicities of entries in Pascal's triangle Operators & calculus links Difference operator
Binomial_type
arrays Lozanić's triangle · Sierpinski triangle · Pascal matrix · Multiplicities of entries in Pascal's triangle Operators & calculus links Difference operator
Alternating_factorial
Transformation of a mathematical sequence
for the transformation, where T is an infinite-dimensional operator with matrix elements Tnk. The transform is an involution, that is, T T = 1 {\displaystyle
Binomial_transform
Theorem in p-adic analysis
arrays Lozanić's triangle · Sierpinski triangle · Pascal matrix · Multiplicities of entries in Pascal's triangle Operators & calculus links Difference operator
Mahler's_theorem
Pascal's triangle. The difference between the two is that an entry in the trinomial triangle is the sum of the three (rather than the two in Pascal's
Trinomial_triangle
arrays Lozanić's triangle · Sierpinski triangle · Pascal matrix · Multiplicities of entries in Pascal's triangle Operators & calculus links Difference operator
Difference_polynomials
Type of prime number
arrays Lozanić's triangle · Sierpinski triangle · Pascal matrix · Multiplicities of entries in Pascal's triangle Operators & calculus links Difference operator
Wilson_prime
Number of orderings allowing ties
doi:10.1137/0405040, MR 1186818 Lewis, Barry (2010), "Revisiting the Pascal matrix", American Mathematical Monthly, 117 (1): 50–66, doi:10.4169/000298910X474989
Ordered_Bell_number
Mathematical identity involving sums of binomial coefficients
arrays Lozanić's triangle · Sierpinski triangle · Pascal matrix · Multiplicities of entries in Pascal's triangle Operators & calculus links Difference operator
Abel's_binomial_theorem
Triangular array of natural numbers
number Delannoy number Motzkin number Narayana polynomials Schröder number Pascal's triangle Learning materials related to Partition related number triangles
Narayana_number
Mathematical sequence
approach tremendously speeds up optimization problems. Stirling numbers Pascal matrix Lah, Ivo (1954). "A new kind of numbers and its application in the actuarial
Lah_number
Arithmetical function
arrays Lozanić's triangle · Sierpinski triangle · Pascal matrix · Multiplicities of entries in Pascal's triangle Operators & calculus links Difference operator
Kempner_function
Mathematical integral
arrays Lozanić's triangle · Sierpinski triangle · Pascal matrix · Multiplicities of entries in Pascal's triangle Operators & calculus links Difference operator
Nørlund–Rice_integral
Integer sequence
after Henry W. Gould that counts how many odd numbers are in each row of Pascal's triangle. It consists only of powers of two, and begins: 1, 2, 2, 4, 2
Gould's_sequence
arrays Lozanić's triangle · Sierpinski triangle · Pascal matrix · Multiplicities of entries in Pascal's triangle Operators & calculus links Difference operator
Egorychev_method
Integer matrices with +1 or −1 determinant; invertible over the integers. GL_n(Z)
mathematics, a unimodular matrix M is a square integer matrix having determinant +1 or −1. Equivalently, it is an integer matrix that is invertible over
Unimodular_matrix
Number that is the product of factorials
arrays Lozanić's triangle · Sierpinski triangle · Pascal matrix · Multiplicities of entries in Pascal's triangle Operators & calculus links Difference operator
Jordan–Pólya_number
= ( n k ) {\displaystyle d_{n,k}={\binom {n}{k}}} , also called the Pascal matrix: P = ( 1 1 1 1 2 1 ⋯ 1 3 3 1 1 4 6 4 1 ⋮ ⋱ ) {\displaystyle
Riordan_array
Mathematics term
(sometimes called Losanitsch's triangle) is a triangular array related to of Pascal's triangle. It is named after the Serbian chemist Sima Lozanić, who researched
Lozanić's_triangle
Data type that represents an ordered collection of elements (values or variables)
mathematical concepts vector and matrix, array types with one and two indices are often called vector type and matrix type, respectively. More generally
Array_(data_type)
Type of number in combinatorial mathematics and statistics
{\displaystyle A_{m}(0,4)=1,4,6,4,1} . If p = 1 {\displaystyle p=1} , Pascal's triangle appears, read along diagonals: A m ( 1 , 1 ) = 1 , 1 , 1 , 1
Fuss–Catalan_number
Philosophical thought experiment about utility
In philosophy, Pascal's mugging is a thought experiment demonstrating a problem in expected utility maximization. A rational agent should choose actions
Pascal's_mugging
Programming language and IDE
language and a software product that uses the Delphi dialect of the Object Pascal programming language and provides an integrated development environment
Delphi_(software)
French actor (born 1958)
Internationally, he is known for playing The Merovingian in The Matrix film series, beginning with The Matrix Reloaded. Wilson is the son of Georges Wilson, who was
Lambert_Wilson
_{k=1}^{n}\left\{{\begin{matrix}n\\k\end{matrix}}\right\}a_{k}} , where { n k } {\displaystyle \left\{{\begin{matrix}n\\k\end{matrix}}\right\}} is the Stirling
Stirling_transform
Mapping function that preserves data point locality
paper. Buluç et al. present a sparse matrix data structure that Z-orders its non-zero elements to enable parallel matrix-vector multiplication. Matrices in
Z-order_curve
French software company
high-tech, consumer products, and medical devices through the acquisition of MatrixOne, which would be linked with ENOVIA. In 2007, subsidiary brand 3DVIA was
Dassault_Systèmes
Expression for sums of powers of integers
resorting to the numbers of Bernoulli but by inverting a matrix easily obtained from the triangle of Pascal. The term Faulhaber polynomials is used by some authors
Faulhaber's_formula
GPU microarchitecture by Nvidia
trademark, for a GPU microarchitecture developed by Nvidia, succeeding Pascal. It was first announced on a roadmap in March 2013, although the first product
Volta_(microarchitecture)
Academic journal
Communication Abstracts Compendex Environment Index GEOBASE Inspec Metadex PASCAL Science Citation Index Expanded Scopus while Part II is covered by Academic
Deep_Sea_Research
Polynomials used for interpolation
phenomenon of large oscillation. Lagrange polynomials are also related to matrix eigenspace projection via Frobenius covariants. Given a set of nodes
Lagrange_polynomial
Computer algebra system
areas of application are multivariate rational function arithmetic and matrix algebra over rings of multivariate polynomials or rational functions. Fermat
Fermat (computer algebra system)
Fermat_(computer_algebra_system)
Characters from post-apocalyptic drama series
The first season, based on 2013's The Last of Us, follows Joel (Pedro Pascal) and Ellie (Bella Ramsey) as they travel across the United States. In the
Characters of The Last of Us (TV series)
Characters_of_The_Last_of_Us_(TV_series)
Numerical computation library for Python
compiler for manipulating and evaluating mathematical expressions, especially matrix-valued ones. In Theano, computations are expressed using a NumPy-esque syntax
Theano_(software)
Hardware acceleration unit for artificial intelligence tasks
low-precision (e.g., FP16, INT8) matrix multiplication operations, they can be used to emulate higher-precision matrix multiplications in scientific computing
Neural_processing_unit
Exponentially decreasing bounds on tail distributions of random variables
for matrix-valued random variables. The following version of the inequality can be found in the work of Tropp. Let M1, ..., Mt be independent matrix valued
Chernoff_bound
Method of data analysis
matrix. In matrix form, the empirical covariance matrix for the original variables can be written 𝐐∝𝐗𝖳𝐗=𝐖𝜦𝐖𝖳 The empirical covariance matrix between
Principal_component_analysis
Feature of GPUs by Nvidia
encoding, and sample adaptive offset (SAO). Nvidia Video Codec SDK 8 added Pascal exclusive Weighted Prediction feature (CUDA based). Weighted prediction
NVENC
Congolese singer-songwriter (1940–2013)
Pascal-Emmanuel Sinamoyi Tabu (13 November 1940 – 30 November 2013), known professionally as Tabu Ley Rochereau, was a Congolese singer-songwriter, record
Tabu_Ley_Rochereau
Description of an algorithm that resembles a computer program
language, although this is discouraged. Some syntax sources include Fortran, Pascal, BASIC, C, C++, Java, Lisp, and ALGOL. Variable declarations are typically
Pseudocode
Mixture of several programming languages in the same program
ALGOL (the origin of the concept), pidgin Fortran, pidgin BASIC, pidgin Pascal, and pidgin C. It is a compact and often informal notation that blends syntax
Pidgin_code
List of programming languages types and the languages that meet its description
NEWP Oberon, Oberon-2 Objective-C OCaml Pascal derivatives Component Pascal Object Pascal Turbo Pascal UCSD Pascal Perl PHP PL/I PureBasic Python R Ruby
List of programming languages by type
List_of_programming_languages_by_type
Diagram that represents a workflow or process
p. 327. ISBN 9780262022255. Goldstine, Herman (1972). The Computer from Pascal to Von Neumann. Princeton University Press. pp. 266–267. ISBN 0-691-08104-2
Flowchart
Academic journal
Contents/Engineering, Computing & Technology EBSCO databases Ei Compendex Inspec PASCAL ProQuest databases Science Citation Index Expanded Scopus According to the
Energy_and_Buildings
Type of image blur produced by a Gaussian function
center point. Values from this distribution are used to build a convolution matrix which is applied to the original image. This convolution process is illustrated
Gaussian_blur
English actor (born 1977)
Black Hawk Down SPC Lance Twombly Credited as Thomas Hardy 2002 Deserter Pascal Dupont Star Trek: Nemesis Shinzon 2003 Dot the i Tom LD 50 Lethal Dose Matt
Tom_Hardy
Jabotinsky matrix (sometimes called iteration matrix or power matrix) is an infinite matrix used to convert function composition into matrix multiplication
Jabotinsky_matrix
Curve from a cone intersecting a plane
matrix}A&B/2&D/2\\B/2&C&E/2\\D/2&E/2&F\end{matrix}}\right)\left({\begin{matrix}x\\y\\z\end{matrix}}\right)=0.} The 3 × 3 matrix above is called
Conic_section
Academic journal
Core Civil Engineering Abstracts Communication Abstracts INSPEC Metadex PASCAL Scopus According to the Journal Citation Reports, the journal has a 2025
Materials Science and Engineering A
Materials_Science_and_Engineering_A
Genus of ornithopod dinosaurs
Victor Feijó de; Marrecas, Pedro; Figueiredo, Silvério D.; Godefroit, Pascal (2025-09-15). "Cariocecus bocagei, a new basal hadrosauroid from the Lower
Haolong
Brand of Nvidia graphics cards used in workstations
Windows 456.38, Linux 455.23.04 for Kepler, Maxwell, Pascal, Volta All Kepler, Maxwell, Pascal, Volta and later can do OpenGL 4.6 with Driver 418+ All
Quadro
Composite material
material made from "pre-impregnated" fibers and a partially cured polymer matrix, such as epoxy or phenolic resin, or even thermoplastic mixed with liquid
Pre-preg
Series of embedded computing boards by Nvidia
scenarios, e.g. for the use in smaller camera drones such as the Skydio 2. A matrix describing a set of performance modes was provided by the media along with
Nvidia_Jetson
Photography software
Retrieved 2012-07-25. Obry, Pascal (2023-06-21). "darktable 4.4.0 released". darktable. Retrieved 2023-09-02. Obry, Pascal (2019-12-24). "darktable 3.0
Darktable
travel, tourism, insurance
PASCAL MATRIX
PASCAL MATRIX
Boy/Male
Australian, French, German
Child of Easter
Boy/Male
Australian, French, German, Hebrew, Latin, Spanish
Born on Easter; Passover
Boy/Male
Bengali, Christian, Danish, French, German, Hebrew, Indian, Italian, Latin, Marathi
Pass over; Child of Easter
Male
Spanish
Spanish form of Latin Paschalis, PASCUAL means "Passover; Easter."
Female
French
Feminine form of French Pascal, PASCALE means "Passover; Easter."
Male
French
Variant spelling of French Pascal, PASCHAL means "Passover; Easter."
Surname or Lastname
English
English : variant of Pascal.
Male
French
French name derived from Latin Paschalis, PASCAL means "Passover; Easter." This name was popular with early Christians, mainly given to sons born at Easter time.
Girl/Female
French
Born at Easter.
Female
Spanish
Feminine form of Spanish Pascual, PASCUALA means "Passover; Easter."
Boy/Male
Hebrew
Born on Passover.
Girl/Female
French
Born at Easter.
Boy/Male
British, Christian, English, Italian
Form of Pascal; Passover
Boy/Male
Muslim
Pasha. A Title.
Boy/Male
French Italian Hebrew Spanish
Born on Easter.
Surname or Lastname
English (Cornwall)
English (Cornwall) : from a Cornish variant of the personal name Pascal.
Boy/Male
Australian, French, Hebrew
Child of Easter; Born on Passover
Boy/Male
French Hebrew Italian
Born on Easter.
Male
Portuguese
Portuguese form of Latin Paschalis, PASCOAL means "Passover; Easter."
Male
Russian
(Паша) Russian pet form of Czech/Russian Pavel, PASHA means "small."
PASCAL MATRIX
PASCAL MATRIX
PASCAL MATRIX
PASCAL MATRIX
PASCAL MATRIX
PASCAL MATRIX
PASCAL MATRIX
n.
State of being a rascal; rascality; domain of rascals; rascals, collectively.
v. t.
To add a parcel or item to; to itemize.
a.
Of or pertaining to the pope of Rome; proceeding from the pope; ordered or pronounced by the pope; as, papal jurisdiction; a papal edict; the papal benediction.
v. t.
To make up into a parcel; as, to parcel a customer's purchases; the machine parcels yarn, wool, etc.
a.
Of or pertaining to the passover, or to Easter; as, a paschal lamb; paschal eggs.
n.
See Pasha.
v. t.
To treat as a vassal; to subject to control; to enslave.
a.
Coming without regularity; occasional; incidental; as, casual expenses.
n.
One of the nasal bones.
a.
Of or pertaining to case; as, a casal ending.
n.
A female rascal.
n.
A public officer in Scotland who prosecutes in petty criminal cases; -- called also procurator fiscal.
n.
A wretch; a rascal.
n.
A part; a portion; a piece; as, a certain piece of land is part and parcel of another piece.
a.
Having a quality imparted by means of the nose; and specifically, made by lowering the soft palate, in some cases with closure of the oral passage, the voice thus issuing (wholly or partially) through the nose, as in the consonants m, n, ng (see Guide to Pronunciation, // 20, 208); characterized by resonance in the nasal passage; as, a nasal vowel; a nasal utterance.
a.
Resembling a vassal; slavish; servile.
a.
Pertaining to, or resembling, a disk; as, discal cells.
a.
Belonging to the post office or mail service; as, postal arrangements; postal authorities.
a.
Having the form of a caecum, or bag with one opening; baglike; as, the caecal extremity of a duct.
n.
Alt. of Pascha
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