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central binomial coefficient∗ rspuzio† 2013-03-21 17:32:49 The nth central binomial coefficient is defined to be 2n (2n)! = n (n!)2 where 2n is a binomial coefficient. These numbers have the generating n function √ 1 = 1 + 2x + 6x2 + 20x3 + 70x4 + 252x5 + · · · 1 − 4x They are closely related to the Catalan sequence, in that 1 2n Cn = n+1 n Alternate definition A less frequently-encountered definition for the nth central binomial coeffi cient is b nn c . 2 Note that the set of these numbers meeting this alternate criterion is a superset of those meeting the first criterion, since for n = 2m we have n 2m 2m = = n 2m b2c b 2 c m By cancelling terms of one of the n!’s against terms of the 2n!, one may rewrite the central binomial coefficient as follows: 2n 2n(2n − 1) · · · (n + 2)(n + 1) = . n n(n − 1) · · · 3 · 2 · 1 ∗ hCentralBinomialCoefficienti created: h2013-03-21i by: hrspuzioi version: h35936i Privacy setting: h1i hDefinitioni h05A10i h11B65i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1 Alternatively, one may cancel each term of the n! against twice itself, leaving 2’s in the numerator: 2n (2n − 1)(2n − 3) · · · 5 · 3 · 1 = 2n n n(n − 1) · · · 3 · 2 · 1 Doubling the terms in the denominator, we obtain an expression for the central binomial coeficient in terms of a quotient of successive odd numbers by successive even numbers: 2n (2n − 1)(2n − 3) · · · 5 · 3 · 1 = 4n n 2n(2n − 2) · · · 6 · 4 · 2 By means of these formulae, one may derive some important properties of the central binomial coeficients. By examining the first two formulae, one may deduce results about the prime factors of central binomial coefficients (for proofs, please see the attachments to this entry): Theorem 1 If n ≥ 3 is an integer and p is a prime number such that n < p < 2n, then p divides 2n n . Theorem 2 If n ≥ 3 is an integer and p is a prime number such that 2n/3 < p ≤ n, then p does not divide 2n n . In conjunction with Wallis’ formula for π, the third formula for the central binomial coefficient may be used to derive an asymptotic expression, as is done in an attachment to this entry: r 2 4n 2n √ ≈ π 2n + 1 n 2