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Central Binomial Coefficient

The $n$th central binomial coefficient is defined as ${n\choose\left\lfloor{n/2}\right\rfloor }$, where ${n\choose k}$ is a Binomial Coefficient and $\left\lfloor{n}\right\rfloor $ is the Floor Function. The first few values are 1, 2, 3, 6, 10, 20, 35, 70, 126, 252, ... (Sloane's A001405). The central binomial coefficients have Generating Function

{1-4x^2-\sqrt{1-4x^2}\over 2(2x^3-x^2)}=1+2x+3x^2+6x^3+10x^4+\ldots.

The central binomial coefficients are Squarefree only for $n=1$, 2, 3, 4, 5, 7, 8, 11, 17, 19, 23, 71, ... (Sloane's A046098), with no others less than 7320.

The above coefficients are a superset of the alternative ``central'' binomial coefficients

{2n\choose n}={(2n)!\over(n!)^2},

which have Generating Function


The first few values are 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, ... (Sloane's A000984).

Erdös and Graham (1980, p. 71) conjectured that the central binomial coefficient ${2n\choose n}$ is never Squarefree for $n>4$, and this is sometimes known as the Erdös Squarefree Conjecture. Sárközy's Theorem (Sárközy 1985) provides a partial solution which states that the Binomial Coefficient ${2n\choose n}$ is never Squarefree for all sufficiently large $n\geq n_0$ (Vardi 1991). Granville and Ramare (1996) proved that the only Squarefree values are $n=2$ and 4. Sander (1992) subsequently showed that ${2n\pm d\choose n}$ are also never Squarefree for sufficiently large $n$ as long as $d$ is not ``too big.''

See also Binomial Coefficient, Central Trinomial Coefficient, Erdös Squarefree Conjecture, Sárközy's Theorem, Quota System


Granville, A. and Ramare, O. ``Explicit Bounds on Exponential Sums and the Scarcity of Squarefree Binomial Coefficients.'' Mathematika 43, 73-107, 1996.

Sander, J. W. ``On Prime Divisors of Binomial Coefficients.'' Bull. London Math. Soc. 24, 140-142, 1992.

Sárközy, A. ``On Divisors of Binomial Coefficients. I.'' J. Number Th. 20, 70-80, 1985.

Sloane, N. J. A. Sequences A046098, A000984/M1645, and A001405/M0769, in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Vardi, I. ``Application to Binomial Coefficients,'' ``Binomial Coefficients,'' ``A Class of Solutions,'' ``Computing Binomial Coefficients,'' and ``Binomials Modulo and Integer.'' §2.2, 4.1, 4.2, 4.3, and 4.4 in Computational Recreations in Mathematica. Redwood City, CA: Addison-Wesley, pp. 25-28 and 63-71, 1991.

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© 1996-9 Eric W. Weisstein