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,

Equivalently, the trinomial coefficients are defined by

The trinomial coefficients also have generating function

The trinomial triangle gives the triangle of trinomial coefficients,

(Sloane's 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 n symbols, each -1, 0, or 1, which sum to k. 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 k = 3, giving

Trinomial coefficients have a rather sparse literature, although no less than Euler (1765) authored a 20-page paper on their properties (Andrews 1990).

The following table gives the first few columns of the trinomial triangle.

The diagonals are summarized in the following table.

The trinomial coefficients satisfy an identity similar to that of the binomial coefficients, namely

(Andrews 1990).

Explicit formulas for are given by

(Andrews 1990), which gives the closed forms
where is a regularized hypergeometric function.

For at least n < 2000 and , is prime iff is prime only for , (3, 0), or (4, 0). It is apparently not known if this property holds for all n. However, the k = 1 column is explicitly given by

where is a Motzkin number, and so can only be prime for n = 2.

Euler (1765) noted the pretty near-identity of central trinomial coefficients

where is a Fibonacci number, which holds only for (Andrews 1990). For n = 0, 1, ..., the first few values of the left side are (Sloane's A103872), while the values on the right side are 0, 2, 2, 6, 12, 30, 72, 182, ... (Sloane's A059727). A couple of pages of Euler's works containing the near-identity 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 q-series identity from which Euler's near result follows. In particular, defining

gives the identities

(Andrews 1990). This then leads to the near-identity via

Andrews (1990) also gave the pretty identities

Binomial Coefficient, Central Trinomial Coefficient, Motzkin Number, Multinomial Coefficient, Trinomial, Trinomial Triangle

search