Generating functions

• How Many Combinations Are Possible with Limited Repetitions?
  Count unordered selections of k items from n types with per-type limits using a generating function or an inclusion-exclusion formula.
• How to Distribute Identical Stones Among Buckets with Limits
  Count distributions of identical stones into distinct buckets with minimum and maximum limits using stars and bars, inclusion-exclusion, or generating functions.
• How to Derive Bessel Function Recurrences from the Generating Function
  Differentiate the Laurent generating function for J_n(x), match coefficients, and obtain the standard order and derivative recurrences.
• How Many Ways Can You Climb n Stairs Using 2 or 3 at a Time?
  Count ways to climb n stairs using only 2- or 3-stair moves: derive the recurrence, generating function, and an explicit binomial-sum formula.
• How to Find a Closed Form for an nth Term Using Generating Functions
  Turn a recurrence into a generating function, simplify it, and extract its coefficients to obtain a closed form for the nth term.