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Topic: Falling factorial

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	Pochhammer symbol - Wikipedia, the free encyclopedia
		and, confusingly, is used in combinatorics to represent the "falling factorial" or "lower factorial"
		are commonly used to denote the rising and falling factorials, respectively.
		The falling factorial occurs in a formula which represents polynomials using the forward difference operator Δ and which is formally similar to Taylor's theorem of calculus.
	en.wikipedia.org /wiki/Pochhammer_symbol (284 words)

	PlanetMath: falling factorial (Site not responding. Last check: 2007-10-08)
		Unfortunately, the notational conventions for the rising and falling factorials lack a common standard, and are plagued with a fundamental inconsistency.
		Works of combinatorics [1,2,3] give greater focus to the falling factorial because of its role in defining the Stirling numbers.
		This is version 8 of falling factorial, born on 2002-02-19, modified 2004-07-06.
	planetmath.org /encyclopedia/PochhammerSymbol.html (322 words)

	Factorial moment - Encyclopedia, History, Geography and Biography (Site not responding. Last check: 2007-10-08)
		In probability theory, the nth factorial moment of a probability distribution, also called the nth factorial moment of any random variable X with that probability distribution, is
		is the falling factorial (confusingly, this same notation, the Pochhammer symbol (x)
		, is used by some mathematicians, especially in the theory of special functions, to denote the rising factorial x(x + 1)(x + 2)...
	www.arikah.net /encyclopedia/Factorial_moment (140 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		] Technique for measurement of liquid densities in which the time of fall of a drop of the sample liquid through a reference liquid is measured.
		] A viscometer which measures the speed of a spherical body falling with constant velocity in the fluid whose viscosity is to be determined.
		The zone or boundary between resistant rocks of older land and weaker strata of plains.
	www.accesscience.com /Dictionary/F/F2/DictF2.html (2479 words)

	Park Physical Therapy Online (Site not responding. Last check: 2007-10-08)
		Besides muscle weakness or joint pain, other contributing factors may be: medications, lack of concentration, memory challenges, poor balance, poor vision, vestibular function, foot problems, diminished sensory feedback, or a loss of proprioception (knowing where your body is in space).
		Falling is just part of old age and cannot be prevented.
		Falls are one of the leading causes of injury of people over 65 living at home.
	www.parkphysicaltherapy.com /falling.html (783 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		Unfortunately, the falling factorial is also often denoted by $(x)_n$, so great care must be taken when encountering this notation.
		{\bf Notes.} Unfortunately, the notational conventions for the rising and falling factorials lack a common standard, and are plagued with a fundamental inconsistency.
		Works focusing on special \htmladdnormallink{functions}{http://planetmath.org/encyclopedia/Function.html} [4,5] universally use $(x)_n$ to denote the rising factorial and use this symbol in the description of the various flavours of hypergeometric series.
	www.ma.utexas.edu /~jcorneli/e/work%20folder/FEM-2004-08-16/TeX/05A10--FallingFactorial.tex (381 words)

	Stirling number
		In combinatorics, Stirling numbers of the second kind S(n,k) (with a capital "S") count the number of equivalence relations having k equivalence classes defined on a set with n elements.
		= 1 because it is an empty product) be the falling factorial, we can characterize the Stirling numbers of the second kind by
		In particular, the nth moment of the Poisson distribution with expected value 1 is precisely the number of partitions of a set of size n, i.e., it is the nth Bell number (this fact is "Dobinski's formula").
	www.ebroadcast.com.au /lookup/encyclopedia/st/Stirling_number.html (134 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		function value = fall (x, n) %% FALL computes the falling factorial function [X]_N.
		% % Moreover, the Stirling numbers of the first kind can be used % to convert a falling factorial into a polynomial, as follows: % % [X]_N = S^0_N + S^1_N * X + S^2_N * X^2 +...
		% If N = 0, FALL = 1, if N = 1, FALL = X. Note that if N is % negative, a "rising" factorial will be computed.
	www.csit.fsu.edu /~burkardt/m_src/subset/fall.m (248 words)

	Symbol
		Phallic symbol Phallic symbols are forms or concepts considered to be representations of the penis (or phallus) and the...
		Pochhammer symbol In mathematics, the Pochhammer symbol for the falling factorial.
		The Schläfli symbol of a polygon with n edges is {n}.
	www.brainyencyclopedia.com /topics/symbol.html (339 words)

	Pochhammer symbol : Falling factorial (Site not responding. Last check: 2007-10-08)
		terms defined : Pochhammer symbol : Falling factorial
		In mathematics, the Pochhammer symbol $(x)_n\,$ is used in the theory of special functions to represent the "rising factorial" or "upperfactorial" $(x)_n=x(x+1)(x+2)\cdots(x+n-1)$ and, confusingly, is used in combinatorics to represent the "falling factorial" or "lower factorial" $(x)_n=x(x-1)(x-2)\cdots(x-n+1).$
		It is to this negligence and Ratisbon: for, had you received my letters regularly, you would have.
	www.termsdefined.net /fa/falling-factorial.html (318 words)

	PlanetMath: Stirling numbers of the first kind (Site not responding. Last check: 2007-10-08)
		are the integer coefficients of the falling factorial polynomials.
		Cross-references: label, singleton, cycle notation, symmetric group, cycles, orbits, permutations, absolute value, summation, equation, recurrence relation, powers, sides, derivative, identities, generating function, binomial formula, connection, initial conditions, formula, terms, characterization, equivalent, observation, relation, polynomials, falling factorial, integer
		This is version 3 of Stirling numbers of the first kind, born on 2002-03-31, modified 2004-11-01.
	planetmath.org /encyclopedia/StirlingNumbers.html (354 words)

	Pochhammer symbol -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		In the following, the notation of (Click link for more info and facts about Ronald L. Graham) Ronald L. Graham, (Click link for more info and facts about Donald E. Knuth) Donald E. Knuth and Oren Patashnik in their book Concrete Mathematics will be adopted.
		Note that the falling factorial can be written as a (Click link for more info and facts about binomial coefficient) binomial coefficient:
		The rising factorial can be generalized to a continuous value of n using the (Click link for more info and facts about Gamma function) Gamma function:
	www.absoluteastronomy.com /encyclopedia/P/Po/Pochhammer_symbol.htm (334 words)

	[No title]
		Input, integer X, the argument of the falling factorial function.
		integer i integer i_factorial integer ierror integer j integer p(n) integer pcopy(n) integer rank !
		integer i integer i_factorial integer j integer k integer jhi integer nperm integer p(n) integer rank integer r1 integer r2 !
	orion.math.iastate.edu /burkardt/f_src/combo/combo.f90 (5864 words)

	Pascal's Triangle - Terminology
		What is a falling factorial, such as 6
		The idea of falling factorial is to start at some value x, which for our purposes can be any real (or even any complex) number and take the product
		, since in the falling factorial if we fall farther than n from n, 0 will be part of the product.
	binomial.csuhayward.edu /Terminology.html (2506 words)

	factorial - OneLook Dictionary Search
		Factorial : Online Plain Text English Dictionary [home, info]
		FACTORIAL : Statistics (in particular, re-randomisation statistics) [home, info]
		Phrases that include factorial: double factorial, factorial design, falling factorial, rising factorial, factorial experiment, more...
	www.onelook.com /cgi-bin/cgiwrap/bware/dofind.cgi?word=factorial (237 words)

	CS111 Section 01: Lab Final (Site not responding. Last check: 2007-10-08)
		The file format will be one number per line in the file, you don't know how many lines there are in the file in advance.
		The output of your program (to the screen) will be the factorial of each of the numbers in the file one per line, and the last line of output will be the sum of all the numbers in the file.
		You must have a Makefile, that compiles your factorial function.c file into a.o file and then correctly links in and compiles your main program.
	www.cs.nmt.edu /~cs111/Labs/Finals/thursdayFinal.html (515 words)

	Rising factorial (Site not responding. Last check: 2007-10-08)
		is used in the theory of special functions to represent the"rising factorial" or "upper factorial"
		and, confusingly, is used in combinatorics to represent the"falling factorial" or "lower factorial"
		The falling factorial occurs in a formula which represents polynomials usingthe forward difference operator Δ; and which is formallysimilar to Taylor's theorem of calculus.
	www.therfcc.org /rising-factorial-329454.html (125 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		] A suitably divided vertical rod, or stick, anchored in an open vessel so that the magnitude of rise and fall of the liquid level may be observed directly.
		A rigid bar hinged to the boom of a dipper or pull shovel and fastened to the bucket.
		n) is asymptotic to factorial n; that is, the limit as n goes to
	www.accesscience.com /Dictionary/S/S54/DictS54.html (2820 words)

	IngentaConnect Calculation of some determinants using the s-shifted factorial (Site not responding. Last check: 2007-10-08)
		The s-shifted factorial is defined as a generalization for non-negative integers of the power function, the rising factorial (or Pochammer's symbol) and the falling factorial.
		It is a special case of a polynomial sequence of the binomial type studied in combinatorics theory.
		They are used to evaluate families of generalized Vandermonde determinants with s-shifted factorials as elements, instead of power functions.
	api.ingentaconnect.com /content/iop/jphysa/2004/00000037/00000022/art00003 (197 words)

	Factorial -- from MathWorld (Site not responding. Last check: 2007-10-08)
		As n grows large, factorials begin acquiring tails of trailing zeros.
		The first few numbers n such that the sum of the factorials of their digits is equal to the
		Conway, J. and Guy, R. "Factorial Numbers." In The Book of Numbers.
	www.massey.ac.nz /~a159202/Factorial%20--%20from%20MathWorld.htm (657 words)

	Factorial (Site not responding. Last check: 2007-10-08)
		Randomised factorial trial of falls prevention among older people living in thei...
		agglutinations.com: 7 Factorial: An Experiment in Writing and Research (Part II)...
		agglutinations.com: 7 Factorial: An Experiment in Writing and Research (Part I)...
	www.scienceoxygen.com /math/98.html (43 words)

	[No title]
		Input, real X, the argument of the falling factorial function.
		Output, real FALL, the value of the falling factorial function.
		fall = 1.0E+00 arg = x if (n > 0) then do i = 1, n fall = fall * arg arg = arg - 1.0E+00 end do else if (n
	www.math.iastate.edu /burkardt/f_src/subset/subset.f90 (5041 words)

	No Title
		The main results of this paper are the three recurrences for these sequences, given uniformly in equations (5), (6), and (7), and the formula of the generation function for factorial moments given in Proposition 5.
		In Section 2 we state these recurrences, which we then determine by generating function methods in Sections 3, 4, and 5.
		21], to one for summing falling factorials of ordinates for paths of E(n,0).
	math.boisestate.edu /~sulanke/COURSE/sulanke/sulanke.html (3630 words)

	SUBSET - Combinatorial Routines
		Interesting number theoretic functions include the Bell, Catalan, Morse-Thue, and pentagonal numbers, rising and falling factorials, and Pythagorean triples.
		dpoly_f2p.m, converts a real polynomial from factorial to standard form.
		dpoly_p2f.m, converts a real polynomial from standard to factorial form.
	www.csit.fsu.edu /~burkardt/m_src/subset/subset.html (2025 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		function value = rise (x, n) %% RISE computes the rising factorial function [X]^N. % Discussion: % % [X]^N = X * (X + 1) * (X + 2) *...
		% % Modified: % % 10 June 2004 % % Author: % % John Burkardt % % Parameters: % % Input, real X, the argument of the rising factorial function.
		% If N = 0, RISE = 1, if N = 1, RISE = X. Note that if N is % negative, a "falling" factorial will be computed.
	www.csit.fsu.edu /~burkardt/m_src/subset/rise.m (277 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		Let p(m) be the probability of getting at least one picture of each player in a sample of size m.
		The raw number of deaths is adjusted to take into account the fact that some hospitals and surgeons take care of sicker patients than others do.
		What is interesting about this article, besides the subheadline "But a few hospitals fall below average", is that a picture is included which shows, for each of the 31 hospitals, the death rate and the 95% confidence intervals.
	www.geom.umn.edu /docs/snell/chance/chance_news/recent_news/chance_news_4.08.html (5571 words)

	[No title]
		Familiar examples are the polynomials in $n$, exponential functions (with an integer base) with exponent a polynomial in $n$, factorials, and the binomial coefficients.
		\eqno(6)$$ Using a generalization of the binomial theorem, it can be shown that $$\sqrt{1-4x}=(1-4x)^{1/2}=1+\sum_{n=1}^{\infty} (-1)^n {{({1\over2})_n}\over{n!}}4^nx^n, \eqno(7)$$ where $({1\over 2})_n$ is the {\sl falling factorial} function $(x)_n=x(x-1)\cdots (x-n+1)$ evaluated at $x=1/2$.
		However, on using the familiar expression for a binomial coefficient involving three factorials, and replacing each of those factorials by an approximate value, we can approximate $c_n$, and use this approximation to find a function $f(n)$ with a relatively simple form such that $c_n=O(f(n))$.
	www.mhhe.com /math/advmath/rosen/texfiles/ch7.tex (8139 words)

	[No title]
		only good for _tiny_ n ulong ffact2num(const ulong fc, ulong n); // Convert (falling) factorial* in fc[] to number.
		// Rising radices: 2, 3,..., n-1 void ffact2cyclic(const ulong fc, ulong n, ulong x); // Generate cyclic permutation (standard representation) in x[] // from the (n-2) digit factorial number in fc[0,...,n-3].
		// Falling radices: n-1,..., 3, 2 // ========== HEADER FILE src/comb/permcyclic.h: ========== class perm_cyclic; // // algorithm of G.G.Langdon Jr., as given in Knuth // // ========== HEADER FILE src/comb/permderange.h: ========== class perm_derange; // ========== HEADER FILE src/comb/permgray.h: ========== class perm_gray; // Minimal change permutations.
	www.jjj.de /fxt/doc/comb-doc.txt (1090 words)

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