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

$${1\over\sqrt{1-4x}}=1+2x+6x^2+20x^3+70x^4+\ldots.$$

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

References

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.'' http://www.research.att.com/~njas/sequences/eisonline.html 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.