Central Trinomial Coefficient
The 
th central trinomial coefficient 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 is also gives the number of permutations of 
symbols, each 
, 0, or 1, which
sum to 0. For example, there are seven such permutations of three symbols: 
, 
, 
, 
, and 
, 
, 
.
The generating function is given by
 |  |  |
(1)
|
 |  |  |
(2)
|
The central trinomial coefficients are given by the recurrence equation
 |
(3)
|
with 
, but cannot be expressed as
a fixed number of hypergeometric terms (Petkovšek et al. 1996, p. 160).
The coefficients satisfy the congruence
 |
(4)
|
(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).
Sum are given by
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
Closed form 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). 
is prime
for 
, 3, and 4, with no others for 
(E. W. Weisstein,
Oct. 30, 2015). It is not known if any other prime central trinomials exist.
Moreover, a more general unproven conjecture states that there are no prime trinomial
coefficients except these three central trinomials and all trinomials of the form
.
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, 
, ... (OEIS A098331),
with closed form
 |
(14)
|
where 
is a Gegenbauer
polynomial. These numbers are prime for 
, 4, 5, 6, 7,
10, 11, 12, 26, 160, 3787, ... (OEIS A112874),
with no others for 
(E. W. Weisstein,
Mar. 7, 2005).
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central binomial coefficient