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Mathematical functions
\end{alignedat}}} Falling and rising factorials of integers are directly related to the ordinary factorial: n ! = 1 ( n ) = ( n ) n , ( m
Falling_and_rising_factorials
Mathematical sequences in combinatorics
express coefficients in expansions of falling and rising factorials as polynomials. That is, the falling factorial, defined as ( x ) n = x ( x − 1 )
Stirling_number
Brackets as used in mathematical notation
The notation ( x ) n {\displaystyle (x)_{n}} is used to denote the falling factorial, an n-th degree polynomial defined by ( x ) n = x ( x − 1 ) ( x −
Bracket_(mathematics)
Expectation or average of the falling factorial of a random variable
the factorial moment is a mathematical quantity defined as the expectation or average of the falling factorial of a random variable. Factorial moments
Factorial_moment
Generalization of the mathematical factorial
Bhargava factorial has the property that many number-theoretic results involving the ordinary factorials remain true even when the factorials are replaced
Bhargava_factorial
In mathematics, a non-algebraic number
coefficients) of factorials j ! {\displaystyle j!} ; in particular P {\displaystyle P} is an integer. Smaller factorials divide larger factorials, so the smallest
Transcendental_number
Type of polynomial sequence
polynomials The Bernoulli polynomials of the second kind The Falling and rising factorials The Touchard polynomials The Mittag-Leffler polynomials Rota
Sheffer_sequence
Generalized Pochhammer symbol q-Pochhammer symbol Pochhammer contour Falling and rising factorials Works by or about Leo August Pochhammer at the Internet Archive
Leo_August_Pochhammer
Term in the mathematical theory of special functions
special cases of the falling and rising factorials, including the Pochhammer symbol, and the generalized cases of the multiple factorial functions (multifactorial
Pochhammer_k-symbol
Permutation List of permutation topics Pochhammer symbol (also falling, lower, rising, upper factorials) Poisson distribution Polygamma function Primorial Proof
List of factorial and binomial topics
List_of_factorial_and_binomial_topics
same as their sum, with the odd-indexed factorials multiplied by −1 if n is even, and the even-indexed factorials multiplied by −1 if n is odd, resulting
Alternating_factorial
Number computed as a product of powers
the 19th century by Hermann Kinkelin and James Whitbread Lee Glaisher. As Kinkelin showed, just as the factorials can be continuously interpolated by the
Hyperfactorial
Numeral system in combinatorics
factorial base, although factorials do not function as base, but as place value of digits. By converting a number less than n! to factorial representation, one
Factorial_number_system
Function defined by a hypergeometric series
used in Falling and rising factorials. Andrews, George E.; Askey, Richard & Roy, Ranjan (1999). Special functions. Encyclopedia of Mathematics and its Applications
Hypergeometric_function
Mathematical sequence
mathematics, the (signed and unsigned) Lah numbers are coefficients expressing rising factorials in terms of falling factorials and vice versa. They were
Lah_number
In probability theory and statistics, the factorial moment generating function (FMGF) of the probability distribution of a real-valued random variable
Factorial moment generating function
Factorial_moment_generating_function
Sequence valued in polynomials
Hermite polynomials Many are studied in algebra and combinatorics: Monomials Rising factorials Falling factorials All-one polynomials Abel polynomials Bell
Polynomial_sequence
Number theory expression
expression for the exponent of the largest power of a prime p that divides the factorial n!. It is named after Adrien-Marie Legendre. It is also sometimes known
Legendre's_formula
Generalization of the binomial theorem to other polynomials
k_{m+1}},} as can easily be seen by writing the three coefficients using factorials as follows: n ! k 1 ! k 2 ! ⋯ k m − 1 ! K ! K ! k m ! k m + 1 ! = n !
Multinomial_theorem
Algebraic expansion of powers of a binomial
the falling factorial. This agrees with the usual definitions when r is a nonnegative integer. Then, if x and y are real numbers with |x| > |y|, and r is
Binomial_theorem
Product of consecutive factorial numbers
{\displaystyle n} factorials. They are a special case of the Jordan–Pólya numbers, which are products of arbitrary collections of factorials. The n {\displaystyle
Superfactorial
Age-related health problem
and quality of life for older adults, and is usually precipitated by multiple risk factors. The cause of falling in old age is often multi-factorial,
Falls_in_older_adults
Arithmetical function
for which S ( n ) {\displaystyle S(n)} is as small as possible are the factorials: S ( k ! ) = k {\displaystyle S(k!)=k} , for all k ≥ 1 {\displaystyle
Kempner_function
Textbook by Ronald Graham, Donald Knuth, and Oren Patashnik
mathematical notation: the Iverson bracket, floor and ceiling functions, and notation for rising and falling factorials. Donald Knuth used the first edition of
Concrete_Mathematics
Type of permutation of a set of elements
subfactorial Dn equals the nearest integer to n!/e, where n! denotes the factorial of n and e ≈ 2.718281828... is Euler's number. The problem of counting derangements
Derangement
Theorem on prime numbers
multiple of n. That is (using the notations of modular arithmetic), the factorial ( n − 1 ) ! = 1 × 2 × 3 × ⋯ × ( n − 1 ) {\displaystyle (n-1)!=1\times
Wilson's_theorem
Mathematical identity involving sums of binomial coefficients
identities Factorials & approximations Factorial · Bhargava factorial · Hyperfactorial · Alternating factorial · Factorial moment · Factorial number system
Abel's_binomial_theorem
Mathematical result on arithmetic properties of binomial coefficients
coefficient in factorial form, using ( a b ) = a ! ( a − b ) ! b ! . {\displaystyle {a \choose b}={\frac {a!}{(a-b)!b!}}.} List of factorial and binomial topics
Star_of_David_theorem
Number that is the product of factorials
orientations of comparability graphs and in the problem of finding factorials that can be represented as products of smaller factorials. The sequence of Jordan–Pólya
Jordan–Pólya_number
Integer sequence
are in each row of Pascal's triangle. It consists only of powers of two, and begins: 1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, 16, 2, 4, ... (sequence
Gould's_sequence
Formula in mathematics
c^{k},} where n is a nonnegative integer and the sum is taken over all combinations of nonnegative indices i, j, and k such that i + j + k = n. The trinomial
Trinomial_expansion
Uniqueness theorem in complex analysis
identically zero, and the finite differences for f uniquely determine its Newton series. That is, if a Newton series for f exists, and the difference satisfies
Carlson's_theorem
Solved prime-number problem
conjectured in 1845 by Joseph Bertrand, it was first proven by Chebyshev, and a shorter but also advanced proof was given by Ramanujan. The following elementary
Proof_of_Bertrand's_postulate
Theorem in p-adic analysis
_{p}} , Mahler's theorem states that f {\displaystyle f} is continuous if and only if its Newton series converges everywhere to f {\displaystyle f} , so
Mahler's_theorem
Triangular array of the binomial coefficients
increases. This can also be seen by applying Stirling's formula to the factorials involved in the formula for combinations. This is related to the operation
Pascal's_triangle
Number of subsets of a given size
object, and the unglued part of the second object.) In this regard, binomial coefficients are to exponential generating series what falling factorials are
Binomial_coefficient
Conjecture in combinatorial number theory
N ( a ) = O ( log a ) . {\displaystyle N(a)=O(\log a).} Abbott, Erdős, and Hanson (1974) (see References) refined the estimate to: N ( a ) = O ( log
Singmaster's_conjecture
Graphical aid for deriving some concepts in combinatorics
In combinatorics, stars and bars (also called sticks and stones, balls and bars, and dots and dividers) is a graphical aid for deriving certain combinatorial
Stars and bars (combinatorics)
Stars_and_bars_(combinatorics)
Mathematical function
the rising factorial (v)n = v(v+1)(v+2) ... (v+n-1), Gn(k) are the Gregory coefficients of higher order with Gn(1) = Gn, Γ is the gamma function and ζ is
Digamma_function
Numbering of combinations of items
(1887). The term "combinadic" is introduced by James McCaffrey. Unlike the factorial number system, the combinatorial number system of degree k is not a mixed
Combinatorial_number_system
Product of the first "n" prime numbers
is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers,
Primorial
Transformation of a mathematical sequence
rising k-binomial transform is sometimes defined as ∑ j = 0 n ( n j ) j k a j . {\displaystyle \sum _{j=0}^{n}{\binom {n}{j}}j^{k}a_{j}.} The falling
Binomial_transform
Type of polynomial sequence
\{x^{n}:n=0,1,2,\ldots \}} is of binomial type. The sequence of "lower factorials" is defined by ( x ) n = x ( x − 1 ) ( x − 2 ) ⋅ ⋯ ⋅ ( x − n + 1 ) . {\displaystyle
Binomial_type
Mathematical theorem on convolved binomial coefficients
{\displaystyle (1+x)^{s+t}} . This identity may be rewritten in terms of the falling Pochhammer symbols as ( s + t ) n = ∑ k = 0 n ( n k ) ( s ) k ( t ) n −
Vandermonde's_identity
Recurrence relations of binomial coefficients in Pascal's triangle
identity on Pascal's triangle: when the addends represented in the summation and the sum itself are highlighted, the shape revealed is vaguely reminiscent
Hockey-stick_identity
-th row are indexed starting with − n {\displaystyle -n} from the left, and the middle entry has index 0. The symmetry of the entries of a row about
Trinomial_triangle
Upper bound on intersecting set families
{\displaystyle (n-1)!!} matchings, where ! ! {\displaystyle !!} denotes the double factorial. The largest family of matchings that pairwise intersect (meaning that
Erdős–Ko–Rado_theorem
Numbers parameterizing ways to partition a set
that combinatorialists use for falling factorials coincides with the notation used in special functions for rising factorials; see Pochhammer symbol. Transformation
Stirling numbers of the second kind
Stirling_numbers_of_the_second_kind
Sequence of numbers ((2n) choose (n))
{\displaystyle {\binom {2\cdot 2}{2}}} is equal to 6, and there are six arrangements of two copies of A and two copies of B: AABB, ABAB, ABBA, BAAB, BABA, BBAA
Central_binomial_coefficient
polynomials, which include the Newton polynomials, Selberg's polynomials, and the Stirling interpolation polynomials as special cases. The general difference
Difference_polynomials
Numerical method for calculating the gamma function
)}^{z}v^{g}\,dv,} and deriving a series for the integral. The following implementation in the Python programming language works for complex arguments and typically
Lanczos_approximation
Family of polynomials
[m-r+1]_{q}}{[1]_{q}[2]_{q}\cdots [r]_{q}}}\quad (r\leq m).} In terms of the q factorial [ n ] q ! = [ 1 ] q [ 2 ] q ⋯ [ n ] q {\displaystyle [n]_{q}!=[1]_{q}[2]_{q}\cdots
Gaussian_binomial_coefficient
Mathematical series
positive integer: where α {\displaystyle \alpha } is any complex number, and the power series on the right-hand side is expressed in terms of the (generalized)
Binomial_series
Mathematical integral
coefficients grow rapidly for large n. Table of Newtonian series List of factorial and binomial topics Niels Erik Nørlund, Vorlesungen uber Differenzenrechnung
Nørlund–Rice_integral
transformation List of factorial and binomial topics Bernstein, M.; Sloane, N. J. A. (1995). "Some canonical sequences of integers". Linear Algebra and Its Applications
Stirling_transform
Branch of discrete mathematics
"Continuous and profinite combinatorics" (PDF). Archived (PDF) from the original on 2009-02-26. Retrieved 2009-01-03. Björner, Anders; Stanley, Richard
Combinatorics
Mathematics term
symmetries exhibited by rows of paraffins (archaic term for alkanes) and isomer types and number of alkanes. First 9 rows of the triangle (sequence A034851
Lozanić's_triangle
Stirling numbers, Bernoulli numbers, Harmonic numbers, Catalan numbers and other combinatorial numbers. The method relies on two observations. First
Egorychev_method
Generalized chain rule in calculus
value of m 1 + m 2 + ⋯ + m n = k {\displaystyle m_{1}+m_{2}+\cdots +m_{n}=k} and noticing that m j {\displaystyle m_{j}} has to be zero for j > n − k + 1
Faà_di_Bruno's_formula
Arrangement of trinomial coefficients
demonstrates the fastest and easiest way to compute the numbers for any layer of the tetrahedron without computing factorials, which quickly become huge
Pascal's_pyramid
Probability distribution
second kind, and n k _ = n ( n − 1 ) ⋯ ( n − k + 1 ) {\displaystyle n^{\underline {k}}=n(n-1)\cdots (n-k+1)} is the k {\displaystyle k} -th falling power of
Binomial_distribution
Infinite matrices with Pascal's triangle as elements
i, j start at 0, and ! denotes the factorial. These matrices along with related variants arise naturally in the study of Hermite and Laguerre polynomials
Pascal_matrix
Generalization of the binomial distribution
various categories. When k is 2 and n is 1, the multinomial distribution is the Bernoulli distribution. When k is 2 and n is bigger than 1, it is the binomial
Multinomial_distribution
Recursive integer sequence
Catalan–Mersenne number Delannoy number Fuss–Catalan number List of factorial and binomial topics Lobb numbers Motzkin number Narayana number Narayana
Catalan_number
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
Ali Akansu, and published in 1990, using the family of binomial polynomials for subband decomposition of discrete-time signals. Akansu and his fellow authors
Binomial_QMF
Count of permutations by cycles
recurrence relation using the definition of Stirling numbers in terms of rising factorials. Distributing the last term of the product, we have x n + 1 ¯ = x
Stirling numbers of the first kind
Stirling_numbers_of_the_first_kind
Personal experiences associated with death or impending death
subjective observations by workers falling from scaffolds, soldiers who suffered injuries, climbers who had fallen from heights and other individuals who had come
Near-death_experience
Triangular array of natural numbers
pairs of parentheses, which are correctly matched (known as Dyck words) and which contain k {\displaystyle k} distinct nestings. For instance, N
Narayana_number
Vehicle propelled fully or mostly by electricity
2018. Retrieved 8 June 2018. "Electric vehicle price is rising, but cost-per-mile is falling". Ars Technica. Archived from the original on 4 June 2018
Electric_vehicle
Systematic classification of 12 related enumerative problems concerning two finite sets
for the third element, and so on. Therefore, instead of by an ordinary power of x, the value is given by a falling factorial power of x, in which each
Twelvefold_way
Sounds and pronunciation of Inuit languages
followed by a falling pitch on the second syllable means "What did you say?" A middle pitch on the first syllable followed by a rising pitch on the second
Inuit_phonology
Study of the analytic properties of scattering amplitudes
{\displaystyle \Gamma (x)} is the gamma function, a generalization of factorial ( x − 1 ) ! {\displaystyle (x-1)!} . This gamma function is a meromorphic
Regge_theory
Natural number
permutations of length 7 without rising or falling successions. 647 is a Chen prime, an Eisenstein prime with no imaginary part, and the sum of five consecutive
600_(number)
Intentional act causing one's own death
J (1970). The Rising Sun: The Decline and Fall of the Japanese Empire, 1936–1945. Random House. p. 519. O'Keeffe TM (1984). "Suicide and Self-Starvation"
Suicide
Historical term in mathematics
symbol used here for the falling sequential product. A similar relationship holds for the backward differences and rising factorial. This series is also known
Umbral_calculus
Family of power series in mathematics
the rising factorial (typically written with a Pochhammer symbol in this context, even though the Pochhammer symbol normally denotes the falling factorial)
Generalized hypergeometric function
Generalized_hypergeometric_function
Discrete-time stochastic process
is the rising factorial and x m _ = ∏ i = 0 m − 1 ( x − i ) {\displaystyle x^{\underline {m}}=\prod _{i=0}^{m-1}(x-i)} is the falling factorial. It is
Chinese_restaurant_process
Plot using the dispersal of scattered dots to show the relationship between variables
example, weight and height would be on the y-axis, and height would be on the x-axis. Correlations may be positive (rising), negative (falling), or null (uncorrelated)
Scatter_plot
Procedure used to randomize a deck of playing cards
when new.) After shuffling, the measure of randomness is the number of rising sequences that are left in each suit. If a computer has access to purely
Shuffling
Mathematical set with repetitions allowed
\over k!},} to match the expression of binomial coefficients using a falling factorial power: ( n k ) = n k _ k ! . {\displaystyle {n \choose k}={n^{\underline
Multiset
Script used to write the Punjabi language
although Punjabi lacks these sounds. Tones in Punjabi can be either rising, neutral, or falling: When the tonal letter is in onset positions, as in the pronunciation
Gurmukhi
Rational number sequence
denotes the rising factorial power in the notation of D. E. Knuth. The numbers βn = Bn/n occur frequently in the study of the zeta function and are significant
Bernoulli_number
is a multi‐factorial event that involves the complex interaction of a number of risk factors. The effect of each individual factor and its importance
Racehorse_injuries
Studies that prepare a person for a specific occupation
qualifications demanded for job entry are rising. This reflects a need for not just a more knowledgeable and skilled workforce, but one that can adapt
Vocational_education
Musical composition method
intervals inverted (so that a rising minor third becomes a falling minor third, or equivalently, a rising major sixth): And the retrograde inversion is
Twelve-tone_technique
Operation on formal power series
generating functions for generalized factorial functions formed as special cases of the generalized rising factorial product functions, or Pochhammer k-symbol
Generating function transformation
Generating_function_transformation
mathematics and logic that defines a function based on the values it takes on smaller arguments, essential for defining functions like factorials and other
Glossary_of_logic
Type of number in combinatorial mathematics and statistics
{mp+r-1}{m-1}}} These variants can be converted into a product, gamma or factorial representations too. Combinatorics Statistics Binomial coefficient Binomial
Fuss–Catalan_number
increasing soil erosion and pest pressures, and thereby driving greater use of land, water and inputs. Rising temperatures and changing weather patterns
Effects of climate change on agriculture
Effects_of_climate_change_on_agriculture
Type of prime number
{\displaystyle (p-1)!+1} , where " ! {\displaystyle !} " denotes the factorial function; compare this with Wilson's theorem, which states that every
Wilson_prime
Analytic function in mathematics
recurrence leads to this other series development that uses the rising factorial and is valid for the entire complex plane ζ ( s ) = s s − 1 − ∑ n = 1
Riemann_zeta_function
20th-century rise in intelligence test scores
draftees in NATO countries in Europe—report raw scores, and those also confirm a trend of rising scores over time. The average rate of increase seems to
Flynn_effect
at UC Riverside and UNLV James D. McCaffrey (B.A. 1975) – software engineer and author; see combinatorial number system and factorial number system Paul
List of University of California, Irvine people
List_of_University_of_California,_Irvine_people
Polynomial sequence
n {\displaystyle (\alpha +1)_{n}} is Pochhammer's symbol (for the rising factorial). In this case, the series for the hypergeometric function is finite
Jacobi_polynomials
Discography of British record label
Catalogue v2.0 [Beta]". Nice, James (2011) [2010]. Shadowplayers: The Rise and Fall of Factory Records (paperback ed.). London: Aurum Press. ISBN 978-1-84513-634-5
Factory_Records_discography
Burning tobacco and inhaling the smoke
Modified Reasons for Smoking Scale: factorial structure, gender effects and relationship with nicotine dependence and smoking cessation in French smokers"
Tobacco_smoking
Use of technology in education to enhance learning and teaching
income level, or class size in the way brick and mortar charter schools are. E-learning also has been rising as a supplement to the traditional classroom
Educational_technology
Economic model for international trade
technologies between countries, mainly focusing on factorial differences such as labor force and resource allocation as to why countries trade with each
Heckscher–Ohlin_model
Education practice
Book: Best Practices, Proven Methodologies, and Lessons Learned. Wiley. ISBN 978-0-7879-7296-7. "Plato Rising". Atarimagazines.com. Retrieved October 24
Blended_learning
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FALLING AND-RISING-FACTORIALS
FALLING AND-RISING-FACTORIALS
FALLING AND-RISING-FACTORIALS
FALLING AND-RISING-FACTORIALS
FALLING AND-RISING-FACTORIALS
FALLING AND-RISING-FACTORIALS
FALLING AND-RISING-FACTORIALS
FALLING AND-RISING-FACTORIALS
FALLING AND-RISING-FACTORIALS
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