Central Binomial Calculator

C(2n,n) = (2n)!/(n!)²

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About Central Binomial Calculator

A central binomial coefficient calculator computing C(2n,n) = (2n)!/(n!)². These are the largest entries in each even row of Pascal's triangle. C(2n,n): 1,2,6,20,70,252,924... Asymptotically 4^n/√(πn). Client-side.

Central Binomial Calculator Features

  • C(2n,n) value
  • Sequence
  • Approximation
  • Pascal row
  • Growth rate
Central binomial coefficients: C(2n,n)=1,2,6,20,70,252,924,3432... They're the middle (largest) entry of even-numbered Pascal's triangle rows. C(2n,n) ~ 4^n/√(πn) by Stirling. Catalan = C(2n,n)/(n+1). Fundamental in combinatorics and analysis.

How to Use

Enter n:

  • C(2n,n): Exact value
  • 4^n/√πn: Approximation
  • Sequence: First values

Properties

Always even for n≥1. Divisible by all primes p with n

Catalan Connection

C_n = C(2n,n)/(n+1). So central binomials are 'raw material' for Catalan numbers. C(2n,n) counts all walks from −n to +n; dividing by n+1 restricts to those staying non-negative (ballot problem).

Step-by-Step Instructions

  1. 1Enter n.
  2. 2Compute C(2n,n).
  3. 3Compare to 4^n/√πn.
  4. 4View sequence.
  5. 5See Catalan link.

Central Binomial Calculator — Frequently Asked Questions

Why are they the largest in each row?+

In row 2n of Pascal's triangle, C(2n,k) is maximized when k=n (the middle). This follows from the ratio C(2n,k)/C(2n,k-1) = (2n-k+1)/k, which is >1 for k<n and <1 for k>n. So the maximum is at k=n.

How good is the 4^n/√πn approximation?+

Very good! For n=10: exact=184756, approx=4^10/√(10π)=184611. Error <0.1%. By Stirling: C(2n,n) = 4^n/√(πn)·(1-1/(8n)+O(1/n²)). The correction terms make it even more precise.

What's the connection to Bertrand's postulate?+

Bertrand's postulate (now theorem): there's always a prime between n and 2n. One proof uses C(2n,n): show it's divisible by any prime p with n<p<2n, then show C(2n,n) can't be a product of 'small' primes alone.

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