Generating function

A formal power series whose coefficients encode a sequence or combinatorial counting data. Algebraic operations on the series can translate into useful relations among the encoded coefficients.

Ordinary generating function: A power series whose coefficient of xⁿ is the nth term of a sequence. This basic form assigns one variable power directly to each sequence index.
Fibonacci sequence: The sequence beginning 0, 1 in which each later term is the sum of the previous two. Its recurrence yields a rational generating function and a closed formula for its terms.
Sequence: An ordered list of values indexed by integers or another ordered set. A generating function begins with a sequence whose entries become its coefficients.
Dirichlet generating function: A series of the form Σ aₙn⁻ˢ that encodes a sequence using powers of integers as exponents. It replaces the ordinary power variable with integer-indexed Dirichlet terms, suiting multiplicative number theory.
Leonhard Euler: An eighteenth-century mathematician whose work shaped number theory, analysis, and combinatorics. Euler used infinite products and series to derive results about integer partitions.
Exponential generating function: A power series whose nth term is multiplied by xⁿ/n! rather than xⁿ. The factorial denominator makes this form especially suited to labeled combinatorial structures.
Partition function (number theory): The function p(n) counting the ways to express n as a sum of positive integers, disregarding order. An infinite product generating function encodes all integer partition counts at once.
Coefficient (algebra): A scalar multiplying a variable or basis element in an algebraic expression. Each coefficient carries the sequence value or counting quantity stored at its index.
Probability-generating function: A power series whose nth coefficient is the probability that a nonnegative integer-valued random variable equals n. It specializes coefficient encoding to probability distributions and their moments.
Abraham de Moivre: An eighteenth-century mathematician known for work in probability, trigonometry, and recurrence sequences. His treatment of recurrence sequences anticipated algebraic methods based on their generating series.

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