gen· er·at· e, e, er·at· gen·

gen· gen· gen· gen· gen· er·at· er·at· er·at· er·at· er·at· e, e,e, e, e,

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gen· er·at· e,

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â€œ

it is a

verb.

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LINE

Generating functions are not functions in the for-

mal sense of a mapping from a domain to a codomain;

the name is merely traditional, and they are sometimes more correctly called generating series.

times more correctly called generating series. the name is merely traditional, and they are somemal sense of a mapping from a domain to a codomain; Generating functions are not functions in the for-

eee

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eee

called generating series.

aditional, and they are some-

g from a domain to a codomain;

-rof eht ni snoitcnuf ton era snoitcnuf gnitar

;niamodoc a ot niamod a morf gnippam a fo esnes

.seires gnitareneg dellac yltcerroc erom s

-emos era yeht dna ,lanoitidart ylerem si eman

are not functions in the for-

NER

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11

In mathematics, a indices to start at 1 generating function is rather than 0), but the a formal power series ease with which they can in one indeterminate, be handled may differ whose coefficients encode considerably. The particinformation about a ular generating function, sequence of numbers if any, that is most useful an that is indexed by in a given context will the natural numbers. depend upon the nature Generating functions of the sequence and the were first introduced by details of the problem Abraham de Moivre in being addressed. 1730, in order to solve the Generating functions are general linear recurrence often expressed in closed In mathematics, a indices to start at 1 problem.[1] One can genform (rather than as a generating function is rather than 0), but the eralize to formal power series), by some expresa formal power series ease with which they can series in more than one sion involving operations in one indeterminate, be handled may differ indeterminate, to encode defined for formal whose coefficients encode considerably. The particinformation about arrays power series. These information about a ular generating function, of numbers indexed by expressions in terms of sequence of numbers if any, that is most useful several natural numbers. the indeterminate x may an that is indexed by in a given context will There are various types involve arithmetic operthe natural numbers. depend upon the nature of generating functions, ations, differentiation Generating functions of the sequence and the including ordinary with respect to x and were first introduced by details of the problem generating functions, composition with (i.e., Abraham de Moivre in being addressed. exponential generating substitution into) other 1730, in order to solve the Generating functions are functions, Lambert generating functions; general linear recurrence often expressed in closed series, Bell series, and Disince these operations problem.[1] One can genform (rather than as a richlet series; definitions are also defined for eralize to formal power series), by some expresand examples are given functions, the result series in more than one sion involving operations below. Every sequence in looks like a function of x. indeterminate, to encode defined for formal principle has a generatIndeed, the closed form information about arrays power series. These ing function of each type expression can often be of numbers indexed by expressions in terms of (except that Lambert and interpreted as a function several natural numbers. the indeterminate x may Dirichlet series require that can be evaluated There are various types involve arithmetic operof generating functions, ations, differentiation including ordinary with respect to x and generating functions, composition with (i.e., exponential generating substitution into) other functions, Lambert generating functions; series, Bell series, and Disince these operations richlet series; definitions are also defined for and examples are given functions, the result below. Every sequence in looks like a function of x. principle has a generatIndeed, the closed form ing function of each type expression can often be (except that Lambert and interpreted as a function Dirichlet series require that can be evaluated

eta·re·neg /tar,enej’/ brev ;etareneg :brev ;setareneg :tneserp nosrep dr3 ;detareneg :esnet tsap ;detareneg :elpicitrap tsap :elpicitrap tneserp ro dnureg gnitareneg gen·er·ate /’jene,rat/ verb verb: generate; 3rd person present: generates; past tense: generated; past participle: generated; gerund or present participle: generating

12

gen·er·ate /’jene,rat/ verb verb: generate; 3rd person present: generates; past tense: generated; past participle: generated; gerund or present participle: generating

2. produce (a set or sequence of items) by performing specified mathematical or logical operations on an initial set.

synonyms:cause, give rise to, lead to, result in, bring about, create, make, produce, engender, spawn, precipitate, prompt, provoke, trigger, spark off, stir up, induce, promote, foster 1. cause (something, esp. an emotion or situation) to arise or come about. â€œchanges that are likely to generate controversyâ€?

13

In mathematics, a indices to start at 1 generating function is rather than 0), but the a formal power series ease with which they can in one indeterminate, be handled may differ whose coefficients encode considerably. The particinformation about a ular generating function, sequence of numbers if any, that is most useful an that is indexed by in a given context will the natural numbers. depend upon the nature Generating functions of the sequence and the were first introduced by details of the problem Abraham de Moivre in being addressed. 1730, in order to solve the Generating functions are general linear recurrence often expressed in closed In mathematics, a indices to start at 1 problem.[1] One can genform (rather than as a generating function is rather than 0), but the eralize to formal power series), by some expresa formal power series ease with which they can series in more than one sion involving operations in one indeterminate, be handled may differ indeterminate, to encode defined for formal whose coefficients encode considerably. The particinformation about arrays power series. These information about a ular generating function, of numbers indexed by expressions in terms of sequence of numbers if any, that is most useful several natural numbers. the indeterminate x may an that is indexed by in a given context will There are various types involve arithmetic operthe natural numbers. depend upon the nature of generating functions, ations, differentiation Generating functions of the sequence and the including ordinary with respect to x and were first introduced by details of the problem generating functions, composition with (i.e., Abraham de Moivre in being addressed. exponential generating substitution into) other 1730, in order to solve the Generating functions are functions, Lambert generating functions; general linear recurrence often expressed in closed series, Bell series, and Disince these operations problem.[1] One can genform (rather than as a richlet series; definitions are also defined for eralize to formal power series), by some expresand examples are given functions, the result series in more than one sion involving operations below. Every sequence in looks like a function of x. indeterminate, to encode defined for formal principle has a generatIndeed, the closed form information about arrays power series. These ing function of each type expression can often be of numbers indexed by expressions in terms of (except that Lambert and interpreted as a function several natural numbers. the indeterminate x may Dirichlet series require that can be evaluated There are various types involve arithmetic operof generating functions, ations, differentiation including ordinary with respect to x and generating functions, composition with (i.e., exponential generating substitution into) other functions, Lambert generating functions; series, Bell series, and Disince these operations richlet series; definitions are also defined for and examples are given functions, the result below. Every sequence in looks like a function of x. principle has a generatIndeed, the closed form ing function of each type expression can often be (except that Lambert and interpreted as a function Dirichlet series require that can be evaluated

{ {

t o

t h a t

g e n e r a t e

a r e

l i k e l y

c o n t r o v e r s y â€?

to bring into existence; cause to be; produce.

â€œ c h a n g e s

In mathematics, a generating function is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers an that is indexed by the natural numbers. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem. One can generalize to formal power series in more than one indeterminate, to encode information about arrays of numbers indexed by several natural numbers.

14

Generating functions are often expressed in closed form (rather than as a series), by so sion involving operations defined for formal power series. These expressions in terms of t minate x may involve arithmetic operations, differentiation with respect to x and compo (i.e., substitution into) other generating functions; since these operations are also define tions, the result looks like a function of x. Indeed, the closed form expression can ofte preted as a function that can be evaluated at (sufficiently small) concrete values of x, and the formal power series as its Taylor series; this explains the designation â€œgenerating fu

(1-)

(2-)

15

(3-)

LINGUISTICS produce (a sentence or other unit,) by the application of a finite set of rules to lexical or other linguistic input.

16

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generate g

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, A generating function is a clothesline on which we hang up a sequence of numbers for display.

3

2 9 4 6 9 3 8 0 9 8 7 3 2 9 8 1 1 2 9 3 8 2 7 9 7 6 6 0 9 4 7 9 7 8 1 9 2 8 8 1 2 7 4 6 1 6 3 4 9 99 77 8 7 8 8 4 1 6 3 6 1 9 7 81 6 3 9 4 7 9 7 8 2 8 8 1 2 4 6 7 6 8 0 9 1 9 9 7 2 8 2 9 3 6 1 7 6 2 9 9 7 8 7 6 1 3 4 2 9 6 9 8 7 3 4 6 9

n I M

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form a line,

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surface, or solid

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a point, line, or surface.

symm etric group

1

2

3

In mathematics, the symmet22 set of symbols to itself. ric group Sn on a finite set of bijective functions from the n symbols is the group whose tations, which are treated as elements are all the permucomposition of such permutations of the n symbols, and whose group operation is the whose group operation is the tations of the n symbols, and composition of such permuelements are all the permutations, which are treated as n symbols is the group whose bijective functions from the ric group Sn on a finite set of

The group operation in a symmetric group is function com-

position, denoted by the symbol ∘ or simply by juxtaposition of

the permutations. The composition f ∘ g of permutations f and g,

1 1 23

The group operation in a symmetric group

F=(13)(45)

( position, denoted by the symbol ∘ or simply by

the permutations. The composition f ∘ g of permu-

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)

GENERATING FUNCTION.

a probability-generating function. ordinary generating function is called of a discrete random variable, then its If an is the probability mass function

24

generate

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generate g

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26

Generating functions are not functions in the formal

sense of a mapping from a domain to a codomain; the

name is merely traditional, and they are sometimes

G(anxn)=âˆ‘ anxn n=0

more correctly called generating series

∞

∑ am, n; xm, ym G(am, n; x, y) = n=0

27

m

am ,

a

n;

x m,

x , n; m

,y

28

gen路 er路at路 e,

29

Polynomials are a special case of ordinary generating functions, corresponding to finite sequenc alently sequences that vanish after a certain point. These are important in that many finite seq usefully be interpreted as generating functions, such as the PoincarĂŠ polynomial, and others.

12345 67890

Polynomials are a special case of ordinary generating functions, corresponding to finite sequences, or equivalently sequences that vanish after a certain point. These are important in that many finite sequences can usefully be interpreted as generating functions, such as the PoincarĂŠ polynomial, and others.

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{

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gen路 er路at路 e,

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gen路 er路at路 e,

3. form (a line, surface, or solid) by notionally moving a point, line, or surface.

2. produce by performing specified mathematical or logical operations on an initial set.

1. cause (something, esp. an emotion or situation) to arise or come about.

{ {

Generate

Published on Jan 19, 2014

gen·er·ate ˈjenəˌrāt/Submit verb 1. cause (something, esp. an emotion or situation) to arise or come about. produce (energy, esp. electricit...

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