Central Binomial Coefficient

The th central binomial coefficient is defined as

			(1)
			(2)

where is a binomial coefficient, is a factorial, and is a double factorial.

These numbers have the generating function

(3)

The first few values are 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, ... (OEIS A000984). The numbers of decimal digits in for , 1, ... are 1, 6, 59, 601, 6019, 60204, 602057, 6020597, ... (OEIS A114501). These digits converge to the digits in the decimal expansion of (OEIS A114493).

The central binomial coefficients are never prime except for .

A scaled form of the central binomial coefficient is known as a Catalan number

(4)

Erdős and Graham (1975) conjectured that the central binomial coefficient is never squarefree for , and this is sometimes known as the Erdős squarefree conjecture. Sárkőzy's theorem (Sárkőzy 1985) provides a partial solution which states that the binomial coefficient is never squarefree for all sufficiently large (Vardi 1991). The conjecture of Erdős and Graham was subsequently proved by Granville and Ramare (1996), who established that the only squarefree values are 2, 6, and 70, corresponding to , 2, and 4. Sander (1992) subsequently showed that are also never squarefree for sufficiently large as long as is not "too big."

The central binomial coefficient is divisible by a prime iff the base- representation of contains no digits greater than (P. Carmody, pers. comm., Sep. 4, 2006). For , the first few such are 1, 3, 4, 9, 10, 12, 13, 27, 28, 30, 31, 36, 37, 39, 40, 81, ... (OEIS A005836).

A plot of the central binomial coefficient in the complex plane is given above.

The central binomial coefficients are given by the integral

(5)

(Moll 2006, Bailey et al. 2007, p. 163).

Using Wolstenholme's theorem and the fact that , it follows that

(6)

for an odd prime (T. D. Noe, pers. comm., Nov. 30, 2005).

A less common alternative definition of the th central binomial coefficient of which the above coefficients are a subset is , where is the floor function. The first few values are 1, 2, 3, 6, 10, 20, 35, 70, 126, 252, ... (OEIS A001405). The central binomial coefficients have generating function

(7)

These modified central binomial coefficients are squarefree only for , 2, 3, 4, 5, 7, 8, 11, 17, 19, 23, 71, ... (OEIS A046098), with no others less than (E. W. Weisstein, Feb. 4, 2004).

A fascinating series of identities involving inverse central binomial coefficients times small powers are given by

	(8)
	(9)
	(10)
	(11)
	(12)

(OEIS A073016, A073010, A086463, and A086464; Comtet 1974, p. 89; Le Lionnais 1983, pp. 29, 30, 41, 36), which follow from the beautiful formula

(13)

for , where is a generalized hypergeometric function. Additional sums of this type include

			(14)
			(15)
			(16)

where is the polygamma function and is the Riemann zeta function (Plouffe 1998).

Similarly, we have

	(17)
	(18)
	(19)
	(20)

(OEIS A086465, A086466, A086467, and A086468; Le Lionnais 1983, p. 35; Guy 1994, p. 257), where is the Riemann zeta function. These follow from the analogous identity

(21)

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