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, ... (Sloane's 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) for , 2, ... are 0, 1, 1, 1, 2, 2, 2, 4, 2, 2, 3, 2, ... (Sloane's A102445). is therefore prime for , 3, and 4, with no others for (E. W. Weisstein, Mar. 14, 2005). It has apparently not been proved that no 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, , ... (Sloane's 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, ... (Sloane's A112874), with no others for (E. W. Weisstein, Mar. 7, 2005).

Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 163, 1974.

Noe, T. D. "On the Divisibility of Generalized Central Trinomial Coefficients." J. Integer Seq. 9, Article 06.2.7, 2006. http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Noe/noe35.html.

Petkovšek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A K Peters, 1996. http://www.cis.upenn.edu/~wilf/AeqB.html.

Sloane, N. J. A. Sequences A002426/M2673, A098331, A102445, and A112874 in "The On-Line Encyclopedia of Integer Sequences."

CITE THIS AS:

Weisstein, Eric W. "Central Trinomial Coefficient." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CentralTrinomialCoefficient.html