Central Binomial Coefficient
The 
th central binomial coefficient is defined as 
, where 
is a BinomialCoefficient and 
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
The central binomial coefficients are Squarefree only for
, 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
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 
is neverSquarefree for 
, and this is sometimes known as the Erdös Squarefree Conjecture. Sárközy's Theorem (Sárközy 1985) provides a partial solution which statesthat the Binomial Coefficient is never Squarefree for all sufficiently large 
(Vardi1991). Granville and Ramare (1996) proved that the only Squarefree values are 
and 4. Sander (1992)subsequently showed that 
are also never Squarefree for sufficiently large as long as 
isnot ``too big.''
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. SequencesA046098,A000984/M1645, andA001405/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.
- Totient Valence Function
- Touchard's Congruence
- Tour
- Tournament
- Tournament Matrix
- Tower of Power
- Towers of Hanoi
- T-Polyomino
- T-Puzzle
- Trace (Complex)
- Trace (Group)
- Trace (Map)
- Trace (Matrix)
- Trace (Tensor)
- Tractory
- Tractrix
- Tractrix Evolute
- Tractrix Radial Curve
- Tractroid
- Transcendental Curve
- Transcendental Equation
- Transcendental Function
- Transcendental Number
- Transcritical Bifurcation
- Transfer Function