The 
th
central trinomial coefficient, for nonnegative
integer 
, is defined as the coefficient
of 
in the expansion of 
. It is therefore the middle column of the trinomial
triangle, i.e., the trinomial coefficient 
.
The first few central trinomial coefficients for 
, 2, ... are 1, 3, 7, 19, 51, 141, 393, ... (OEIS A002426).
The central trinomial coefficient also gives the number of ordered sequences of 
symbols, each 
, 0, or 1, which sum to 0. For example,
there are seven such sequences of three symbols: 
,
,
,
,
,
,
and 
.
Writing 
,
the ordinary generating function is given
by
 |  |  |
(1)
|
 |  |  |
(2)
|
The central trinomial coefficients are given by the recurrence equation
 |
(3)
|
for 
,
with 
,
but the sequence cannot be expressed as a sum of a fixed number (independent of 
)
of hypergeometric terms (Petkovšek
et al. 1996, p. 160).
The coefficients satisfy the congruence
 |
(4)
|
for 
(T. D. Noe, pers. comm., Mar. 15, 2005) and
 |
(5)
|
for 
a prime, which is easy to show using Fermat's
little theorem (T. D. Noe, pers. comm., Oct. 26, 2005).
Sums are given by
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
where 
is the floor function. Closed
forms include
 |  |  |
(9)
|
 |  |  |
(10)
|
 |  |  |
(11)
|
 |  |  |
(12)
|
 |  |  |
(13)
|
where 
is a Gegenbauer polynomial, 
is a Legendre polynomial,
and 
is a regularized hypergeometric
function.
The numbers of prime factors (with multiplicity) of 
for 
,
2, ... are 0, 1, 1, 1, 2, 2, 2, 4, 2, 2, 3, 2, ... (OEIS A102445).
E. W. Weisstein (Oct. 30, 2015) reported that 
is prime only for 
,
3, and 4 in the range 
. It is not known if any other prime
central trinomials exist. The more general conjecture states that a prime trinomial coefficient 
occurs only at 
, 
, or 
, or when 
and 
is prime.
A plot of the central trinomial coefficient in the complex plane is given above.
Considering instead the coefficient of 
in the expansion of 
for 
, 2, ... gives the corresponding sequence 
, 
, 5, 
, 
, 41, 
, 
, 365, 
, ..., equal to 
times OEIS A098331,
with closed form
 |
(14)
|
where 
is a Gegenbauer polynomial. According to
OEIS A112874, their absolute
values have been proved prime for 
, 4, 5, 6, 7, 10, 11, 12, 26, 160, and 3787, whereas 
gives a probable prime. The reported search found
no other indices with a prime or probable
prime absolute value for 
(E. W. Weisstein, Mar. 7, 2005).
