\end{alignedat}}} Falling and rising factorials of integers are directly related to the ordinary factorial: n ! = 1 ( n ) = ( n ) n , ( m...
19 KB (3,377 words) - 01:53, 5 April 2025
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 −...
13 KB (1,824 words) - 08:05, 1 June 2025
Stirling number (category Factorial and binomial topics)
in expansions of falling and rising factorials (also known as the Pochhammer symbol) as polynomials. That is, the falling factorial, defined as ( x...
29 KB (4,065 words) - 08:57, 3 June 2025
coefficients) of factorials j ! {\displaystyle j!} ; in particular P {\displaystyle P} is an integer. Smaller factorials divide larger factorials, so the smallest...
52 KB (6,818 words) - 21:52, 1 July 2025
Permutation List of permutation topics Pochhammer symbol (also falling, lower, rising, upper factorials) Poisson distribution Polygamma function Primorial Proof...
2 KB (220 words) - 05:14, 5 March 2025
Pochhammer k-symbol (category Factorial and binomial topics)
special cases of the falling and rising factorials, including the Pochhammer symbol, and the generalized cases of the multiple factorial functions (multifactorial...
8 KB (1,732 words) - 00:26, 23 May 2025
Sheffer sequence (category Factorial and binomial topics)
polynomials; The Bernoulli polynomials of the second kind; The Falling and rising factorials; The Touchard polynomials; The Mittag-Leffler polynomials; Rota...
7 KB (1,049 words) - 19:28, 20 June 2025
Hypergeometric function (category Factorial and binomial topics)
used in Falling and rising factorials. Andrews, George E.; Askey, Richard & Roy, Ranjan (1999). Special functions. Encyclopedia of Mathematics and its Applications...
38 KB (6,920 words) - 21:17, 14 April 2025
Generalized Pochhammer symbol q-Pochhammer symbol Pochhammer contour Falling and rising factorials Works by or about Leo August Pochhammer at the Internet Archive...
3 KB (348 words) - 16:21, 9 December 2024
Lah number (category Factorial and binomial topics)
mathematics, the (signed and unsigned) Lah numbers are coefficients expressing rising factorials in terms of falling factorials and vice versa. They were...
11 KB (1,812 words) - 15:40, 30 October 2024
Polynomial sequence (category Sequences and series)
Hermite polynomials Many are studied in algebra and combinatorics: Monomials Rising factorials Falling factorials All-one polynomials Abel polynomials Bell...
2 KB (176 words) - 15:36, 14 August 2021
Concrete Mathematics (section Contents and history)
mathematical notation: the Iverson bracket, floor and ceiling functions, and notation for rising and falling factorials. Donald Knuth used the first edition of...
6 KB (470 words) - 20:33, 28 November 2024
Stirling numbers of the second kind (category Factorial and binomial topics)
that combinatorialists use for falling factorials coincides with the notation used in special functions for rising factorials; see Pochhammer symbol. Transformation...
25 KB (4,328 words) - 18:26, 20 April 2025
These numbers represent coefficients expressing rising factorials in terms of falling factorials, and vice-versa; equivalently in this case to the number...
14 KB (2,004 words) - 14:56, 9 April 2025
Binomial coefficient (category Factorial and binomial topics)
object, and the unglued part of the second object.) In this regard, binomial coefficients are to exponential generating series what falling factorials are...
62 KB (10,790 words) - 13:49, 8 July 2025
notation to represent the rising factorial.) Suppose X has a Poisson distribution with expected value λ, then its factorial moment generating function...
3 KB (396 words) - 21:10, 14 April 2025
subjective observations by workers falling from scaffolds, soldiers who suffered injuries, climbers who had fallen from heights and other individuals who had come...
87 KB (9,811 words) - 23:07, 3 July 2025
600 (number) (redirect from Six hundred and four)
an n X n board, with 1 in each row and column. Also number of permutations of length n without rising or falling successions)". The On-Line Encyclopedia...
24 KB (3,965 words) - 13:08, 27 June 2025
Suicide (redirect from Suicide and evolution)
Toland, The Rising Sun: The Decline and Fall of the Japanese Empire 1936–1945, Random House, 1970, p. 519 O'Keeffe TM (1984). "Suicide and Self-Starvation"...
182 KB (18,746 words) - 11:50, 3 July 2025
Twelvefold way (redirect from Permutations and Combinations)
fewer 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...
43 KB (5,609 words) - 19:20, 19 January 2025
Stirling numbers of the first kind (category Factorial and binomial topics)
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...
38 KB (7,265 words) - 04:38, 9 June 2025
Chinese restaurant process (section Relationship between Dirichlet-categorical and one-parameter CRP)
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...
24 KB (3,990 words) - 10:53, 6 December 2024
Flynn effect (category Race and intelligence controversy)
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...
64 KB (7,695 words) - 13:53, 9 July 2025
Scatter plot (category Statistical charts and diagrams)
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)...
12 KB (1,332 words) - 13:51, 9 June 2025
Riemann zeta function (category Zeta and L-functions)
1; that context gives rise to a series expansion in terms of the falling factorial. On the basis of Weierstrass's factorization theorem, Hadamard gave...
74 KB (10,718 words) - 01:21, 7 July 2025
Racehorse injuries (redirect from Welfare and Safety of the Racehorse)
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...
77 KB (7,316 words) - 14:21, 9 July 2025
Multiset (category Factorial and binomial topics)
\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...
35 KB (4,983 words) - 20:48, 3 July 2025
Binomial transform (category Factorial and binomial topics)
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...
14 KB (2,438 words) - 15:02, 19 April 2025
Vocational education (redirect from Technical and vocational education and training)
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...
116 KB (14,288 words) - 14:29, 9 July 2025
Regge theory (section History and implications)
{\displaystyle \Gamma (x)} is the gamma function, a generalization of factorial ( x − 1 ) ! {\displaystyle (x-1)!} . This gamma function is a meromorphic...
13 KB (1,919 words) - 06:29, 23 February 2025