Central Binomial Coefficient

The th central binomial coefficient, for a nonnegative integer , is defined as

			(1)
			(2)

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

These numbers have the generating function, convergent for ,

(3)

For , 2, ..., 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). If is this digit count, then converges to (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)

The Erdős squarefree conjecture asserts that the central binomial coefficient is never squarefree for . Sárközy's theorem (Sárközy 1985, Vardi 1991) establishes this for all sufficiently large . Granville and Ramaré (1996) proved the conjecture for all . For positive integers , the only squarefree values are 2, 6, and 70, corresponding to , 2, and 4; gives the additional squarefree value 1. Sander (1992) showed more generally that, for any fixed , is not squarefree for all sufficiently large and all nonnegative integers .

For a nonnegative integer and a prime , the central binomial coefficient is not divisible by iff every digit in the base- representation of is less than (Erdős et al. 1975, p. 84; P. Carmody, pers. comm., Sep. 4, 2006). For , the first few positive 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 a nonnegative integer and is the floor function. For , 2, ..., the first few values are 1, 2, 3, 6, 10, 20, 35, 70, 126, 252, ... (OEIS A001405). Their shifted generating function, , converges for and is

(7)

The displayed expression is assigned its limit 1 at . Among positive integers , these modified central binomial coefficients are squarefree precisely for , 2, 3, 4, 5, 7, 8, 11, 17, 19, 23, and 71 (OEIS A046098; E. W. Weisstein, Feb. 4, 2004). The value at is also squarefree.

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

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

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

(12)

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

			(13)
			(14)
			(15)

where is the polygamma function and is the Riemann zeta function (Plouffe 1998; Borwein et al. 2001, p. 27).

Similarly, we have

	(16)
	(17)
	(18)
	(19)

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

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