A trinomial coefficient is a coefficient of the trinomial triangle. Following the notation of Andrews (1990), the trinomial coefficient , with and , is given by the coefficient of in the expansion of . Therefore,
(1)

The trinomial coefficient can be given by the closed form
(2)

where is a Gegenbauer polynomial.
Equivalently, the trinomial coefficients are defined by
(3)

The trinomial coefficients also have generating function
(4)
 
(5)

i.e.,
(6)

The trinomial triangle gives the triangle of trinomial coefficients,
(7)

(OEIS A027907).
The central column of the trinomial triangle gives the central trinomial coefficients.
The trinomial coefficient is also given by the number of permutations of symbols, each , 0, or 1, which sum to . For example, there are seven permutations of three symbols which sum to 0, , , , , and , , , so .
An alternative (but different) definition of the trinomial coefficients is as the coefficients in (Andrews 1990), which is therefore a multinomial coefficient with , giving
(8)

Trinomial coefficients have a rather sparse literature, although no less than Euler (in 1765) authored a 20page paper on their properties (Andrews 1990).
The following table gives the first few columns of the trinomial triangle.
OEIS  trinomial coefficients  
0  A002426  1, 1, 3, 7, 19, 51, 141, 393, 1107, 3139, 8953, ... 
1  A005717  1, 2, 6, 16, 45, 126, 357, 1016, 2907, 8350, ... 
2  A014531  1, 3, 10, 30, 90, 266, 784, 2304, ... 
3  A014532  1, 4, 15, 50, 161, 504, 1554, 4740, ... 
4  A014533  1, 5, 21, 77, 266, 882, 2850, 9042, ... 
5  A098470  1, 6, 28, 112, 414, 1452, 4917, ... 
The diagonals are summarized in the following table.
The trinomial coefficients satisfy an identity similar to that of the binomial coefficients, namely
(9)

(Andrews 1990).
Explicit formulas for are given by
(10)
 
(11)

(Andrews 1990), which gives the closed forms
(12)
 
(13)

where is a regularized hypergeometric function.
For at least and , is prime iff is prime (since in that case ) or , (3, 0), or (4, 0). It is apparently not known if this property holds for all . However, the column is explicitly given by
(14)

where is a Motzkin number, and so can only be prime for .
In 1765, Euler noted the pretty nearidentity of central trinomial coefficients
(15)

where is a Fibonacci number, which holds only for (Andrews 1990). For , 1, ..., the first few values of the left side are (OEIS A103872), while the values on the right side are 0, 2, 2, 6, 12, 30, 72, 182, ... (OEIS A059727). A couple of pages of Euler's works containing the nearidentity was sent by D. Knuth to R. K. Guy, who included it in his article on the strong law of small numbers (Guy 1990). Meanwhile, Guy had met G. Andrews at the Bateman retirement conference and showed it to him. Half an hour later, Andrews came back with the series identity from which Euler's near result follows. In particular, defining
(16)

gives the identities
(17)
 
(18)
 
(19)
 
(20)
 
(21)
 
(22)
 
(23)
 
(24)
 
(25)
 
(26)
 
(27)
 
(28)
 
(29)
 
(30)
 
(31)

(Andrews 1990). This then leads to the nearidentity via
(32)
 
(33)
 
(34)

Andrews (1990) also gave the pretty identities
(35)
