Trinomial Coefficient

A trinomial coefficient is a coefficient of the trinomial triangle. Following the notation of Andrews (1990), the trinomial coefficient (n; k)_2, with n>=0 and -n<=k<=n, is given by the coefficient of x^(n+k) in the expansion of (1+x+x^2)^n. Therefore,

 (n; -k)_2=(n; k)_2.

The trinomial coefficient can be given by the closed form

 (n; k)_2={1   for n=k=0; C_(k+n)^((-n))(-1/2)   otherwise,

where C_n^((lambda))(z) is a Gegenbauer polynomial.

Equivalently, the trinomial coefficients are defined by

 (1+x+x^(-1))^n=sum_(k=-n)^n(n; k)_2x^k.

The trinomial coefficients also have generating function



 sum_(n=0)^inftysum_(k=-n)^n(n; -k)_2z^nw^(k+n)=1/(1-z(1+w+w^2)).

The trinomial triangle gives the triangle of trinomial coefficients,

1   1   1
1   2   3   2   1
1   3   6   7   6   3   1
1  4  10  16  19  16  10  4  1

(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 n symbols, each -1, 0, or 1, which sum to k. For example, there are seven permutations of three symbols which sum to 0, {-1,0,1}, {-1,1,0}, {0,-1,1}, {0,0,0}, and {0,1,-1}, {1,-1,0}, {1,0,-1}, so (3; 0)_2=7.

An alternative (but different) definition of the trinomial coefficients is as the coefficients in (x+y+z)^n (Andrews 1990), which is therefore a multinomial coefficient with k=3, giving


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

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

kOEIS(n,k)-trinomial coefficients
0A0024261, 1, 3, 7, 19, 51, 141, 393, 1107, 3139, 8953, ...
1A0057171, 2, 6, 16, 45, 126, 357, 1016, 2907, 8350, ...
2A0145311, 3, 10, 30, 90, 266, 784, 2304, ...
3A0145321, 4, 15, 50, 161, 504, 1554, 4740, ...
4A0145331, 5, 21, 77, 266, 882, 2850, 9042, ...
5A0984701, 6, 28, 112, 414, 1452, 4917, ...

The diagonals (n; n-k)_2 are summarized in the following table.

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

 (n; k)_2=(n-1; k-1)_2+(n-1; k)_2+(n-1; k+1)_2

(Andrews 1990).

Explicit formulas for (n; k)_2 are given by

(n; k)_2=sum_(j=0)^(n)(n!)/(j!(j+k)!(n-2j-k)!)
(n; k)_2=sum_(j=0)^(n)(-1)^j(n; j)(2n-2j; n-k-j)

(Andrews 1990), which gives the closed forms

(n; k)_2=(n!)/((n-k)!)_2F^~_1(1/2(k-n),1/2(k-n+1);k+1;4)

where _2F^~_1(a,b;c;z) is a regularized hypergeometric function.

For at least n<2000 and k<=n, (n; k)_2 is prime iff n=k+1 is prime (since in that case (n; k)_2=n) or (n,k)=(2,0), (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

 (n; 1)_2=nM_(n-1),

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

In 1765, Euler noted the pretty near-identity of central trinomial coefficients

 3(n+1; 0)_2-(n+2; 0)_2=F_n(F_n+1),

where F_n is a Fibonacci number, which holds only for n<=7 (Andrews 1990). For n=0, 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 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

 E_n(a,b)=sum_(k=-infty)^infty[(n; 10k+a)_2-(n; 10k+b)_2]

gives the identities


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

3(n+1; 0)_2-(n+2; 0)_2=2(n+1; 0)_2-2(n+1; 1)_2
=2E_(n+1)(0,1) for n<=7
=F_n(F_n+1) for n<=7.

Andrews (1990) also gave the pretty identities

 sum_(k=-infty)^infty(n; 10k+a)_2={1/2F_n^2+1/2F_(n-1)^2+1/2F_(n-1)F_n+2/5F_(n-1)+1/5F_n+1/(10)3^n   for a=0; 1/2F_(n-1)F_nF_n+1/2F_n^2-1/(10)F_(n-1)+1/5F_n+1/(10)3^n   for a=1; 1/2F_(n-1)F_n-1/(10)F_(n-1)-3/(10)F_n+1/(10)3^n   for a=2; -1/2F_(n-1)F_n-1/(10)F_(n-1)-3/(10)F_n+1/(10)3^n   for a=3; -1/2F_(n-1)F_n-1/2F_n^2-1/(10)F_(n-1)+1/5F_n+1/(10)3^n   for a=4; -1/2F_(n-1)^2-1/2F_(n-1)F_n-1/2F_n^2+2/5F_(n-1)+1/5F_n+1/(10)3^n   for a=5.

See also

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

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Andrews, G. "Euler's 'exemplum memorabile inductionis fallacis' and q-Trinomial Coefficients." J. Amer. Math. Soc. 3, 653-669, 1990.Andrews, G. and Baxter, R. J. "Lattice Gas Generalization of the Hard Hexagon Model. III. q-Trinomial Coefficients." J. Stat. Phys. 47, 297-330, 1987.Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 78, 1974.Hoggatt, V. E. Jr., and Bicknell, M. "Diagonal Sums of Generalized Pascal Triangles." Fib. Quart. 7, 341-358 and 393, 1969.Euler, L. "Exemplum Memorabile Inductionis Fallacis." Opera Omnia, Series Prima, Vol. 15. Leipzig, Germany: Teubner, 50-69, 1911.Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science. Reading, MA: Addison-Wesley, p. 575, 1990.Guy, R. K. "The Second Strong Law of Small Numbers." Math. Mag. 63, 3-20, 1990.Henrici, P. Applied and Computational Complex Analysis, Vol. 1: Power Series-Integration-Conformal Mapping-Location of Zeros. New York: Wiley, p. 42, 1988.Riordan, J. Combinatorial Identities. New York: Wiley, p. 74, 1979.Shapiro, L. W.; Getu, S.; Woan, W.-J.; and Woodson, L. C. "The Riordan Group." Disc. Appl. Math. 34, 229-239, 1991.Sloane, N. J. A. Sequences A000027, A000217, A000574, A002426/M2673, A005581, A005712, A005714, A005715, A005716, A005717/M1612, A014531, A014532, A014533, A027907, A059727, A098470, and A103872 in "The On-Line Encyclopedia of Integer Sequences."Warnaar, S. O. "q-Trinomial Identities." 5 Oct. 1998.

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Trinomial Coefficient

Cite this as:

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

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