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Transformation of a mathematical sequence
In combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that computes its forward differences. It is closely
Binomial_transform
Formal power series
{z}{(1-z)^{2}}}\right)} (see also the binomial transform and the Stirling transform). There are also integral formulas for converting between
Generating_function
inverse transform leads to the generating function identity f ( x ) = g ( log ( 1 + x ) ) {\displaystyle f(x)=g(\log(1+x))} . Binomial transform Generating
Stirling_transform
transform Shehu transform Stieltjes transformation Wavelet transform (integral) Weierstrass transform Binomial transform Discrete Fourier transform,
List_of_transforms
Mathematical operator acting on sequences
examples for sequence transformations include the binomial transform, Möbius transform, and Stirling transform. For a given sequence ( s n ) n ∈ N , {\displaystyle
Sequence_transformation
supported orthonormal wavelet transform perspective in 1988 (Daubechies wavelet). It was an extension of Akansu's prior work on binomial coefficient and Hermite
Binomial_QMF
Probability distribution
In probability theory and statistics, the negative binomial distribution, also called a Pascal distribution, is a discrete probability distribution that
Negative binomial distribution
Negative_binomial_distribution
Triangular array of the binomial coefficients
mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics
Pascal's_triangle
Probability distribution
In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes
Binomial_distribution
Algebraic expansion of powers of a binomial
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem
Binomial_theorem
Number of subsets of a given size
mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is
Binomial_coefficient
Mathematical series
In mathematics, the binomial series is a generalization of the binomial formula to cases where the exponent is not a positive integer: where α {\displaystyle
Binomial_series
filters) Binomial series Binomial theorem Binomial transform Binomial type Carlson's theorem Catalan number Fuss–Catalan number Central binomial coefficient
List of factorial and binomial topics
List_of_factorial_and_binomial_topics
Rational number sequence
OEIS: A051714/OEIS: A051715. An autosequence is a sequence which has its inverse binomial transform equal to the signed sequence. If the main diagonal is zeroes = OEIS: A000004
Bernoulli_number
Family of polynomials
mathematics, the Gaussian binomial coefficients (also called Gaussian coefficients, Gaussian numbers, Gaussian polynomials, or q-binomial coefficients) are q-analogs
Gaussian_binomial_coefficient
Type of polynomial sequence
which the index of each polynomial equals its degree, is said to be of binomial type if it satisfies the sequence of identities pn(x+y)=∑k=0n(nk)pk(x)pn−k(y)
Binomial_type
Sequence of numbers ((2n) choose (n))
In mathematics the nth central binomial coefficient is the particular binomial coefficient ( 2 n n ) = ( 2 n ) ! ( n ! ) 2 for all n ≥ 0. {\displaystyle
Central_binomial_coefficient
Graphical aid for deriving some concepts in combinatorics
distinguishable bins. The solution to this particular problem is given by the binomial coefficient (n+k−1k−1), which is the number of subsets of size k − 1 that
Stars and bars (combinatorics)
Stars_and_bars_(combinatorics)
Type of permutation of a set of elements
Vandermonde's identity · Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion
Derangement
Square matrix in which each ascending skew-diagonal from left to right is constant
B_{n}} is the Hankel transform of the sequence b k . {\displaystyle b_{k}.} The Hankel transform is invariant under the binomial transform of a sequence. That
Hankel_matrix
Mathematical operation
{F}}f(e^{-x})\right\}(-is).} The Mellin transform also connects the Newton series or binomial transform to the Poisson generating function by means
Mellin_transform
Recursive integer sequence
ballot theorem Binomial transform Catalan's triangle Catalan–Mersenne number Delannoy number Fuss–Catalan number List of factorial and binomial topics Lobb
Catalan_number
Operation on formal power series
Transform". MathWorld. Solution to exercise 5.71 in Concrete Mathematics. Spivey, M. Z. (2006). "The k-binomial transforms and the Hankel transform"
Generating function transformation
Generating_function_transformation
Mathematical integral
Nørlund–Rice integral to the Mellin transform is not accidental, but is related by means of the binomial transform and the Newton series. In this cycle
Nørlund–Rice_integral
Product of consecutive factorial numbers
Vandermonde's identity · Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion
Superfactorial
Statistical confidence interval for success counts
In statistics, a binomial proportion confidence interval is a confidence interval for the probability of success calculated from the outcome of a series
Binomial proportion confidence interval
Binomial_proportion_confidence_interval
Conjecture in combinatorial number theory
prime numbers appear two times; 6 appears three times, as do all central binomial coefficients except for 1 and 2; (it is in principle not excluded that
Singmaster's_conjecture
Branch of discrete mathematics
astronomer Rabbi Abraham ibn Ezra (c. 1140) established the symmetry of binomial coefficients, while a closed formula was obtained later by the talmudist
Combinatorics
Number computed as a product of powers
Vandermonde's identity · Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion
Hyperfactorial
Theorem on prime numbers
Vandermonde's identity · Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion
Wilson's_theorem
Type of permutation
OEIS: A016116(n) yields: 1. The first column is OEIS: A122045. Its binomial transform leads to: The first row of this array is OEIS: A155585. The absolute
Alternating_permutation
Mathematical theorem on convolved binomial coefficients
identity (or Vandermonde's convolution) is the following identity for binomial coefficients: (m+nr)=∑k=0r(mk)(nr−k) for any nonnegative integers r, m
Vandermonde's_identity
Recurrence relations of binomial coefficients in Pascal's triangle
+X^{n}={\frac {X^{r}-X^{n+1}}{1-X}}={\frac {X^{n+1}-X^{r}}{x}}} . Further, by the binomial theorem, we also find that X r + k = ( 1 + x ) r + k = ∑ i = 0 r + k (
Hockey-stick_identity
Mathematical identity involving sums of binomial coefficients
Abel's binomial theorem, named after Niels Henrik Abel, is a mathematical identity involving sums of binomial coefficients. It states the following: ∑
Abel's_binomial_theorem
Generalization of the binomial theorem to other polynomials
of the terms in that sum. It is the generalization of the binomial theorem from binomials to multinomials. For any positive integer m and any non-negative
Multinomial_theorem
Number theory expression
Vandermonde's identity · Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion
Legendre's_formula
Function in discrete mathematics
In mathematics, the discrete Fourier transform (DFT) is a discrete version of the Fourier transform that converts a finite sequence of numbers into another
Discrete_Fourier_transform
Numerical method for calculating the gamma function
Vandermonde's identity · Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion
Lanczos_approximation
Mathematical function
integral for the gamma function. The integral may be recognized as a Laplace transform. Binet's second integral for the gamma function gives a different formula
Digamma_function
Mathematical functions
{\displaystyle (x)_{n}} with yet another meaning, namely to denote the binomial coefficient ( x n ) {\displaystyle {\tbinom {x}{n}}} . In this article
Falling_and_rising_factorials
Numbering of combinations of items
argument proves that every N can be written in exactly one way as a sum of k binomial coefficients of the given form. The given formula allows finding the place
Combinatorial_number_system
Arrangement of trinomial coefficients
triangle, which contains the binomial coefficients that appear in the binomial expansion and the binomial distribution. The binomial and trinomial coefficients
Pascal's_pyramid
Solved prime-number problem
mathematical publications. The basic idea is to show that the central binomial coefficients must have a prime factor within the interval (n,2n) in order
Proof_of_Bertrand's_postulate
Generalization of the binomial distribution
probability theory, the multinomial distribution is a generalization of the binomial distribution. For example, it models the probability of counts for each
Multinomial_distribution
Data structure that acts as a priority queue
In computer science, a binomial heap is a data structure that acts as a priority queue. It is an example of a mergeable heap (also called meldable heap)
Binomial_heap
Theorem in p-adic analysis
{\displaystyle n} th binomial coefficient polynomial. Here, the n {\displaystyle n} th forward difference is computed by the binomial transform, so that ( Δ n
Mahler's_theorem
Expectation or average of the falling factorial of a random variable
involve Stirling numbers of the second kind. If a random variable X has a binomial distribution with success probability p ∈ [0,1] and number of trials n
Factorial_moment
Natural number
Retrieved 22 May 2022. Sloane, N. J. A. (ed.). "Sequence A007317 (Binomial transform of Catalan numbers)". The On-Line Encyclopedia of Integer Sequences
700_(number)
Upper bound on intersecting set families
the number of sets in A {\displaystyle {\mathcal {A}}} is at most the binomial coefficient ( n − 1 r − 1 ) . {\displaystyle {\binom {n-1}{r-1}}.} The
Erdős–Ko–Rado_theorem
Mapping involving integration between function spaces
In mathematics, an integral transform is a type of transformation that maps a function from its original function space into another function space via
Integral_transform
Recursive mathematical formula
Vandermonde's identity · Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion
Exponential_factorial
Mathematical operation that predicts future values of a discrete-time signal
{\displaystyle a_{i}} are given by the corresponding row of the triangle of binomial transform coefficients. This estimate might be suitable for a slowly varying
Linear_prediction
Generalization of the mathematical factorial
Vandermonde's identity · Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion
Bhargava_factorial
Formula in mathematics
tetrahedron. The trinomial expansion can be calculated by applying the binomial expansion twice, setting d = b + c {\displaystyle d=b+c} , which leads
Trinomial_expansion
Mathematical sequences in combinatorics
of Stirling number. The notation of brackets and braces, in analogy to binomial coefficients, was introduced in 1935 by Jovan Karamata and promoted later
Stirling_number
Generalized chain rule in calculus
Vandermonde's identity · Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion
Faà_di_Bruno's_formula
{\displaystyle p} pairs of identical cards from the two sets, which is the binomial coefficient ( n p ) {\displaystyle {n \choose p}} . The remaining k − 2
Trinomial_triangle
Mathematical result on arithmetic properties of binomial coefficients
arithmetic properties of binomial coefficients. It was discovered by Henry W. Gould in 1972. The greatest common divisors of the binomial coefficients forming
Star_of_David_theorem
Uniqueness theorem in complex analysis
\choose n}\,\Delta ^{n}f(0)} where ( z n ) {\textstyle {z \choose n}} is the binomial coefficient and Δ n f ( 0 ) {\displaystyle \Delta ^{n}f(0)} is the n-th
Carlson's_theorem
Orthogonal wavelets
Akansu, R.A. Haddad and H. Caglar, Perfect Reconstruction Binomial QMF-Wavelet Transform, Proc. SPIE Visual Communications and Image Processing, pp.
Daubechies_wavelet
Numeral system in combinatorics
integer quotient, until this quotient becomes 0. For example, 46310 can be transformed into a factorial representation by these successive divisions: The process
Factorial_number_system
techniques introduced by Georgy Egorychev for finding identities among sums of binomial coefficients, Stirling numbers, Bernoulli numbers, Harmonic numbers, Catalan
Egorychev_method
Regression analysis technique
In statistics, binomial regression is a regression analysis technique in which the response (often referred to as Y) has a binomial distribution: it is
Binomial_regression
Discrete analog of a derivative
Forward differences applied to a sequence are sometimes called the binomial transform of the sequence, and have a number of interesting combinatorial properties
Finite_difference
Probability distribution
In probability theory and statistics, the Poisson binomial distribution is the discrete probability distribution of a sum of independent Bernoulli trials
Poisson_binomial_distribution
Vandermonde's identity · Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion
Alternating_factorial
n-1} \choose {n-1}}} where ( z n ) {\displaystyle {z \choose n}} is the binomial coefficient. For β = 0 {\displaystyle \beta =0} , the generated polynomials
Difference_polynomials
Discrete probability distribution
Poisson distribution. The Poisson distribution is also the limit of a binomial distribution, for which the probability of success for each trial is p=λn
Poisson_distribution
Summation method for some divergent series
equal to or close to −1/z) this series converges to 1/1 − z. Binomial transform Borel summation Cesàro summation Lambert summation Perron's formula
Euler_summation
Product of the first "n" prime numbers
and oscillate infinitely around e {\displaystyle e} later on. Since the binomial coefficient ( 2 n n ) {\displaystyle {\tbinom {2n}{n}}} is divisible by
Primorial
k}\sum _{j=0}^{k}(-1)^{k-j}{k \choose j}f(a+jh).} Binomial transform List of factorial and binomial topics Nörlund–Rice integral Carlson's theorem Davis
Table_of_Newtonian_series
Mathematical technique used in data compression and analysis
mathematical definition of an orthonormal wavelet and of the integral wavelet transform. A function ψ ∈ L 2 ( R ) {\displaystyle \psi \,\in \,L^{2}(\mathbb {R}
Wavelet_transform
Infinite matrices with Pascal's triangle as elements
combinatorics, a Pascal matrix is a matrix (possibly infinite) containing the binomial coefficients as its elements. It is thus an encoding of Pascal's triangle
Pascal_matrix
Integer sequence
(starting from n = 0) gives the highest power of 2 that divides the central binomial coefficient ( 2 n n ) {\displaystyle {\tbinom {2n}{n}}} , and it gives
Gould's_sequence
Mathematics term
Vandermonde's identity · Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion
Lozanić's_triangle
Statistical sequence characterizing probability distributions
coefficients of the rth L-moment are the same as in the rth term of the binomial transform, as used in the r-order finite difference (finite analog to the derivative)
L-moment
Function in statistics
distribution. More abstractly, the logit is the natural parameter for the binomial distribution . The logit function is the negative of the derivative of
Logit
Statistical concept
statistics, the Anscombe transform, named after Francis Anscombe, is a variance-stabilizing transformation that transforms a random variable with a Poisson
Anscombe_transform
Statistical transformation
as |ρ| is not too large and N is not too small. The behavior of this transform has been extensively studied since Fisher introduced it in 1915. Fisher
Fisher_transformation
Square (0,1) matrix
coefficients include the Mobius inversion formula, the binomial transform, and the Stirling transform, among others. Redheffer star product Odlyzko, A. M
Redheffer_matrix
Transform in numerical harmonic analysis
discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet transforms, a key advantage
Discrete_wavelet_transform
Type of prime number
Vandermonde's identity · Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion
Wilson_prime
Special point process in probability
{\displaystyle \xi (A)\sim \operatorname {Bin} (n,P(A)).} The Laplace transform of a binomial process is given by L P , n ( f ) = [ ∫ exp ( − f ( x ) ) P (
Binomial_process
Mathematical sequence
n , k ) {\displaystyle L(n,k)} are given by the formula involving the binomial coefficient L ( n , k ) = ( n − 1 k − 1 ) n ! k ! {\displaystyle L(n,k)={n-1
Lah_number
Method of data analysis
analysis, visualization and data preprocessing. The data are linearly transformed onto a new coordinate system such that the directions (principal components)
Principal_component_analysis
Arithmetical function
Vandermonde's identity · Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion
Kempner_function
Type of number in combinatorial mathematics and statistics
means that they are related to the binomial coefficient. The key difference between Fuss-Catalan and the binomial coefficient is that there are no "illegal"
Fuss–Catalan_number
Infinite sequence of numbers satisfying a linear equation
polynomial), with coefficients given by the corresponding element of the binomial transform. The first few such equations are s n = 1 ⋅ s n − 1 {\displaystyle
Constant-recursive_sequence
Triangular array of natural numbers
Narayana (1930–1987). The Narayana numbers can be expressed in terms of binomial coefficients: N(n,k)=1k(n−1k−1)(nk−1)=1n(nk)(nk−1) The first eight rows
Narayana_number
Probability distribution
conjugate prior probability distribution for the Bernoulli, binomial, negative binomial, and geometric distributions. The formulation of the beta distribution
Beta_distribution
Statistical model for count data
log-linear model, especially when used to model contingency tables. Negative binomial regression is a popular generalization of Poisson regression because it
Poisson_regression
related transforms: Continuous wavelet transform (CWT) Discrete wavelet transform (DWT) Multiresolution analysis (MRA) Lifting scheme Binomial QMF (BQMF)
List of wavelet-related transforms
List_of_wavelet-related_transforms
Turkish-American mathematician (born 1958)
linear subspace methods including sub-band and wavelet transforms, particularly the binomial QMF (also known as Daubechies wavelet) and the multivariate
Ali_Akansu
Number that is the product of factorials
Vandermonde's identity · Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion
Jordan–Pólya_number
Signal representation
domains with a pair of mathematical operators called transforms. An example is the Fourier transform, which converts a time function into a complex valued
Frequency_domain
Near-field diffraction
{\rho ^{2}+z^{2}}}=z{\sqrt {1+{\frac {\rho ^{2}}{z^{2}}}}}.} Next, by the binomial expansion, 1 + u = ( 1 + u ) 1 2 = 1 + u 2 − u 2 8 + ⋯ {\displaystyle {\sqrt
Fresnel_diffraction
actuarial science, the Esscher transform (Gerber & Shiu 1994) is a transform that takes a probability density f(x) and transforms it to a new probability density
Esscher_transform
after Niels Henrik Abel (1802–1829), a Norwegian mathematician. Abel's binomial theorem Abel elliptic functions Abel equation Abel equation of the first
List of things named after Niels Henrik Abel
List_of_things_named_after_Niels_Henrik_Abel
Generalization of the product rule in calculus
( n − k ) ! {\displaystyle {n \choose k}={n! \over k!(n-k)!}} is the binomial coefficient and f ( j ) {\displaystyle f^{(j)}} denotes the j-th derivative
General_Leibniz_rule
after Pierre-Simon Laplace de Moivre-Laplace theorem that approximates binomial distribution with a normal distribution Laplace–Bayes estimator Laplace
List of things named after Pierre-Simon Laplace
List_of_things_named_after_Pierre-Simon_Laplace
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