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Everything about Binomial coefficient

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In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written or ⁠⁠. It is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n; this coefficient can be computed by the multiplicative formula Wikipedia →

🧭 Start here: a 6-step reading path

  1. Read the overview: Binomial coefficientA 5-minute orientation
  2. Skim a recent paper: Integral representations, monotonic properties, and inequalities on extended central binomial coefficientInfluential work since 2022
  3. Pick up a classic book: Art of Proving Binomial IdentitiesThe most-reprinted book on the subject
  4. Read the landmark paper: Interesting Series Involving the Central Binomial CoefficientCited 116 times
  5. Go sideways: Stirling numbers of the first kindA closely related topic
  6. Go sideways: Falling and rising factorialsA closely related topic

📖 Definitions

Noun a coefficient of any of the terms in the expansion of the binomial (x+y), defined by (nk)=n!k!(n−k)!, read as "n choose k"

🗂️ The subject at a glance

  1. History and notation
  2. Definition and interpretations
  3. Computing the value of binomial coefficients
  4. Pascal's triangle
  5. Combinatorics and statistics
  6. Binomial coefficients as polynomials
  7. Identities involving binomial coefficients
  8. Generating functions
  9. Divisibility properties
  10. Bounds and asymptotic formulas
  11. Generalizations

🏆 Landmark papers (most cited)

🆕 Influential since 2022

📚 Books worth reading

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