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, ... (OEIS 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)
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(2)
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The central trinomial coefficients are given by the recurrence equation
(3)
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with , but cannot be expressed as a fixed number of hypergeometric terms (Petkovšek et al. 1996, p. 160).
The coefficients satisfy the congruence
(4)
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(T. D. Noe, pers. comm., Mar. 15, 2005) and
(5)
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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)
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(7)
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(8)
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Closed form include
(9)
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(10)
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(11)
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(12)
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(13)
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where is a Gegenbauer polynomial, is a Legendre polynomial, and is a regularized hypergeometric function.
The numbers of prime factors (with multiplicity) of for , 2, ... are 0, 1, 1, 1, 2, 2, 2, 4, 2, 2, 3, 2, ... (OEIS A102445). is prime for , 3, and 4, with no others for (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 .
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, , ... (OEIS A098331), with closed form
(14)
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where is a Gegenbauer polynomial. These numbers are prime for , 4, 5, 6, 7, 10, 11, 12, 26, 160, 3787, ... (OEIS A112874), with no others for (E. W. Weisstein, Mar. 7, 2005).