OGF Operations

Combinatorics · Axiom Academy

Learn how operations on generating functions translate to operations on sequences When we add two generating functions, we add their corresponding sequence coefficients term-by-term. Multiplying generating functions produces a convolution of the sequences. The coefficient of x^n in the product is the sum of all products a_i · b_j where i + j = n. Taking the derivative of an OGF multiplies each coefficient by its index and shifts indices down by one. Integrating an OGF divides each coefficient by its new index (after shifting up by one). The most important operation: extracting the coefficient of x^n from a generating function to find the n-th term of the sequence.

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