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The 45 pages that link to Generating function, each with the reason it gives.

• Binomial coefficientRelated: The polynomial \((1+x)^n\) encodes the coefficients across a fixed row.

• CombinatoricsRelated: Algebraic operations on these series can represent combinatorial constructions.

• Power seriesRelated: Power-series coefficients can package sequences and reveal their properties.

• SequenceRelated: It packages an entire sequence into an algebraic object.

• Recurrence relationRelated: Generating functions convert recurrence constraints into algebraic equations.

• Fibonacci sequenceRelated: The Fibonacci numbers are encoded by the rational function x/(1−x−x²).

• Radius of convergenceRelated: Its radius often signals the growth rate of the encoded sequence.

• Identity theoremRelated: Equality of generating functions near the origin can establish equality throughout a connected region of analyticity.

• Formal power seriesRelated: Formal series let counting identities work without convergence assumptions.

• CoefficientRelated: Reading a sequence from its generating function is precisely reading its coefficients.

• Pascal's triangleRelated: The rows and diagonals can be studied through power-series identities that generate their coefficients.

• Additive number theoryRelated: Products of generating functions encode the ways summands combine to form an integer.

• Cauchy productRelated: Products of generating functions combine sequences through the Cauchy-product coefficient rule.

• Legendre polynomialsRelated: The Legendre generating function is (1−2xt+t²)^−1/2, with Pℓ(x) as its coefficients.

• Difference equationRelated: Transforming a sequence into a power series can turn its recurrence into an algebraic equation.

• Stars and barsRelated: Its product form derives the same distribution counts algebraically.

• Integer sequenceRelated: Its coefficients package the terms into an algebraic object.

• Lucas sequenceRelated: Lucas sequences have rational generating functions derived directly from their initial values and recurrence.

• Multinomial coefficientRelated: Multinomial coefficients occur as coefficients in powers of sums and related series.

• Multinomial theoremRelated: Generating functions recover multinomial coefficients by multiplying repeated sums.

• Pólya enumeration theoremRelated: Substituting a color inventory into the cycle index produces the desired counts.

• Newton's identitiesRelated: Logarithmic derivatives of a root product yield the power sums and identities.

• Pascal's ruleRelated: The binomial generating function offers an algebraic setting for coefficient recurrences.

• Discrete mathematicsRelated: Generating functions turn discrete counting questions into algebraic operations.

• Jacobi's four-square theoremRelated: Its coefficient at n is the number of four-square representations of n.

• The Art of Computer ProgrammingRelated: Generating functions let the books derive exact counts for combinatorial algorithms.

• Abel's binomial theoremRelated: Generating functions provide an alternative framework for proving binomial identities.

• Algebraic combinatoricsRelated: Packages counting data into an algebraic object that can be manipulated.

• Analytic functionRelated: Analytic properties of generating functions reveal coefficient growth and asymptotics.

• Askey–Gasper inequalityRelated: Generating functions organize polynomial coefficients and support identities used in sum estimates.

• Combinatorial principlesRelated: Multiplication of generating functions implements product-style counting.

• Series expansionRelated: Series coefficients can package discrete data and expose relations among its terms.

• Squared triangular numberRelated: A generating function provides an alternative algebraic encoding of the sequence.