In mathematics the *n*th **central binomial coefficient** is defined in terms of the binomial coefficient by

(
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)

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 *I*_{0} 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 *C*_{n} 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.