Central binomial coefficient ( $(n 2 n)$ )

The central binomial coefficient is the largest coefficient in the expansion of

(1 + 1)^{2 n}

:

(n 2 n) = \frac{( 2 n )!}{( n ! ) ^{2}} .

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- Normalized central binomial coefficients Central binomial coefficient

Normalized central binomial coefficients ( $c_{n} = 4^{- n} (n 2 n)$ )

c_{n} = 4^{- n} (n 2 n)

, with

c_{0} = 1

. Cancellation of factorials gives

c_{n} = \prod_{j = 1}^{n} (1 - \frac{1}{2 j}), \frac{c _{n + 1}}{c _{n}} = 1 - \frac{1}{2 n + 2} .

Consequently

c_{n}

decreases to zero:

lo g c_{n} \leq - \frac{1}{2} \sum_{j = 1}^{n} 1/ j \leq - \frac{1}{2} lo g (n + 1)

. The inequality uses

lo g (1 - x) \leq - x

and comparison with

\int_{1}^{n + 1} d t / t

. Conversely

c_{n} \geq 1/ (n + 1)

by induction, since its ratio is at least

(n + 1) / (n + 2)

. Thus

\sum c_{n}

diverges while

\sum (- 1)^{n} c_{n}

has conditional convergence by the alternating series test. These elementary bounds require no asymptotic formula for factorials.

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Central binomial coefficient by

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The central binomial coefficient is a specific binomial coefficient that appears in combinatorial mathematics, particularly in counting problems and polynomial expansions.

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