Harman Patil (Editor)

Central binomial coefficient

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In mathematics the nth central binomial coefficient is defined in terms of the binomial coefficient by

Contents

( 2 n n ) = ( 2 n ) ! ( n ! ) 2  for all  n ≥ 0.

They are called central since they show up exactly in the middle of the even-numbered rows in Pascal's triangle. The first few central binomial coefficients starting at n = 0 are:

1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, … (sequence A000984 in the OEIS)

Properties

The central binomial coefficients have ordinary generating function

1 1 − 4 x = 1 + 2 x + 6 x 2 + 20 x 3 + 70 x 4 + 252 x 5 + ⋯

and exponential generating function

∑ n = 0 ∞ ( 2 n n ) x n n ! = e 2 x I 0 ( 2 x ) ,

where I0 is a modified Bessel function of the first kind.

The Wallis product can be written in asymptotic form for the central binomial coefficient:

( 2 n n ) ∼ 4 n π n .

The latter can also be easily established by means of Stirling's formula. On the other hand, it can also be used as a means to determine the constant 2 π in front of the Stirling formula, by comparison.

Simple bounds are given by

4 n 2 n + 1 ≤ ( 2 n n ) ≤ 4 n  for all  n ≥ 1

Some better bounds are

4 n 4 n ≤ ( 2 n n ) ≤ 4 n 3 n + 1  for all  n ≥ 1

and, if more accuracy is required,

( 2 n n ) = 4 n π n ( 1 − c n n )  where  1 9 < c n < 1 8 for all n ≥ 1.

The only central binomial coefficient that is odd is 1.

The closely related Catalan numbers Cn are given by:

C n = 1 n + 1 ( 2 n n ) = ( 2 n n ) − ( 2 n n + 1 )  for all  n ≥ 0.

A slight generalization of central binomial coefficients is to take them as Γ ( 2 n + 1 ) Γ ( n + 1 ) 2 = 1 n B ( n + 1 , n ) , with appropriate real numbers n, where Γ ( x ) is the Gamma function and B ( x , y ) is the Beta function.

The powers of two that divide the central binomial coefficients are given by Gould's sequence.

References

Central binomial coefficient Wikipedia
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