Central Trinomial Coefficient

The nth central trinomial coefficient is defined as the coefficient of x^n in the expansion of (1+x+x^2)^n. It is therefore the middle column of the trinomial triangle, i.e., the trinomial coefficient (n; 0)_2. The first few central trinomial coefficients for n=1, 2, ... are 1, 3, 7, 19, 51, 141, 393, ... (OEIS A002426).

The central trinomial coefficient is also gives the number of permutations of n symbols, each -1, 0, or 1, which sum to 0. For example, there are seven such permutations of three symbols: {-1,0,1}, {-1,1,0}, {0,-1,1}, {0,0,0}, and {0,1,-1}, {1,-1,0}, {1,0,-1}.

The generating function is given by


The central trinomial coefficients are given by the recurrence equation


with a_0=a_1=1, but cannot be expressed as a fixed number of hypergeometric terms (Petkovšek et al. 1996, p. 160).

The coefficients satisfy the congruence

 (n+1; 0)_2=(n; 0)_2 (mod n)

(T. D. Noe, pers. comm., Mar. 15, 2005) and

 (p; 0)_2=1 (mod p)

for p a prime, which is easy to show using Fermat's little theorem (T. D. Noe, pers. comm., Oct. 26, 2005).

Sum are given by

(n; 0)_2=sum_(k=0)^(n)(n; 2k)(2k; k)
=sum_(k=0)^(n)(-1)^k(n; j)(2n-2k; n-k).

Closed form include

(n; 0)_2=_2F_1(1/2(1-n),-1/2n;1;4)

where C_n^((lambda))(x) is a Gegenbauer polynomial, P_n(x) is a Legendre polynomial, and _2F^~_1(a,b;c;z) is a regularized hypergeometric function.

The numbers of prime factors (with multiplicity) of (n; 0)_2 for n=1, 2, ... are 0, 1, 1, 1, 2, 2, 2, 4, 2, 2, 3, 2, ... (OEIS A102445). (n; 0)_2 is prime for n=2, 3, and 4, with no others for n<=203661 (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 (n; n-1)_2.


A plot of the central trinomial coefficient in the complex plane is given above.

Considering instead the coefficient of x^n in the expansion of (x^2-x-1)^n for n=1, 2, ... gives the corresponding sequence -1, -1, 5, -5, -11, 41, -29, -125, 365, -131, ... (OEIS A098331), with closed form


where C_n^((lambda))(x) is a Gegenbauer polynomial. These numbers are prime for n=3, 4, 5, 6, 7, 10, 11, 12, 26, 160, 3787, ... (OEIS A112874), with no others for n<10^5 (E. W. Weisstein, Mar. 7, 2005).

See also

Central Binomial Coefficient, Central Fibonomial Coefficient, Trinomial Coefficient

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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.šek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A K Peters, 1996., N. J. A. Sequences A002426/M2673, A098331, A102445, and A112874 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Central Trinomial Coefficient

Cite this as:

Weisstein, Eric W. "Central Trinomial Coefficient." From MathWorld--A Wolfram Web Resource.

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