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Central Binomial Coefficient


The nth central binomial coefficient is defined as

(2n; n)=((2n)!)/((n!)^2)
(1)
=(2^n(2n-1)!!)/(n!),
(2)

where (n; k) is a binomial coefficient, n! is a factorial, and n!! is a double factorial.

These numbers have the generating function

 1/(sqrt(1-4x))=1+2x+6x^2+20x^3+70x^4+....
(3)

The first few values are 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, ... (OEIS A000984). The numbers of decimal digits in (2·10^n; 10^n) for n=0, 1, ... are 1, 6, 59, 601, 6019, 60204, 602057, 6020597, ... (OEIS A114501). These digits converge to the digits in the decimal expansion of log_(10)4=0.6020599... (OEIS A114493).

The central binomial coefficients are never prime except for n=1.

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

 C_n=1/(n+1)(2n; n).
(4)

Erdős and Graham (1975) conjectured that the central binomial coefficient (2n; n) is never squarefree for n>4, 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 (2n; n) is never squarefree for all sufficiently large n>=n_0 (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 n=1, 2, and 4. Sander (1992) subsequently showed that (2n+/-d; n) are also never squarefree for sufficiently large n as long as d is not "too big."

The central binomial coefficient (2n; n) is divisible by a prime p iff the base-p representation of n contains no digits greater than p/2 (P. Carmody, pers. comm., Sep. 4, 2006). For p=3, the first few such n are 1, 3, 4, 9, 10, 12, 13, 27, 28, 30, 31, 36, 37, 39, 40, 81, ... (OEIS A005836).

CentralBinomialCoefficientReIm
CentralBinomialCoefficientContours

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

The central binomial coefficients are given by the integral

 (2n; n)=(2^(2n+1))/piint_0^infty(dx)/((x^2+1)^(n+1))
(5)

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

Using Wolstenholme's theorem and the fact that 2(2p-1; p-1)=(2p; p), it follows that

 (2p; p)=2 (mod p^3)
(6)

for p>3 an odd prime (T. D. Noe, pers. comm., Nov. 30, 2005).

A less common alternative definition of the nth central binomial coefficient of which the above coefficients are a subset is (n; |_n/2_|), where |_n_| 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

 (1-4x^2-sqrt(1-4x^2))/(2(2x^3-x^2))=1+2x+3x^2+6x^3+10x^4+....
(7)

These modified central binomial coefficients are squarefree only for n=1, 2, 3, 4, 5, 7, 8, 11, 17, 19, 23, 71, ... (OEIS A046098), with no others less than 10^6 (E. W. Weisstein, Feb. 4, 2004).

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

sum_(n=1)^(infty)1/((2n; n))=1/(27)(2pisqrt(3)+9)=0.7363998587...
(8)
sum_(n=1)^(infty)1/(n(2n; n))=1/9pisqrt(3)=0.6045997881...
(9)
sum_(n=1)^(infty)1/(n^2(2n; n))=1/3zeta(2)=1/(18)pi^2=0.5483113556...
(10)
sum_(n=1)^(infty)1/(n^4(2n; n))=(17)/(36)zeta(4)=(17)/(3240)pi^4=0.5110970825...
(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

 sum_(n=1)^infty1/(n^k(2n; n))=1/2_(k+1)F_k(1,...,1_()_(k+1);3/2,2,...,2_()_(k-1);1/4)
(13)

for k>=1, where _mF_n(a_1,...,a_m;b_1,...,b_n;x) is a generalized hypergeometric function. Additional sums of this type include

sum_(n=1)^(infty)1/(n^3(2n; n))=1/(18)pisqrt(3)[psi_1(1/3)-psi_1(2/3)]-4/3zeta(3)
(14)
sum_(n=1)^(infty)1/(n^5(2n; n))=1/(432)pisqrt(3)[psi_3(1/3)-psi_3(2/3)]-(19)/3zeta(5)+1/9zeta(3)pi^2
(15)
sum_(n=1)^(infty)1/(n^7(2n; n))=(11)/(311040)pisqrt(3)[psi_5(1/3)-psi_5(2/3)]-(493)/(24)zeta(7)+1/3zeta(5)pi^2+(17)/(1620)zeta(3)pi^4,
(16)

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

Similarly, we have

sum_(n=1)^(infty)((-1)^(n-1))/((2n; n))=1/(25)[5+4sqrt(5)csch^(-1)(2)]=0.3721635763...
(17)
sum_(n=1)^(infty)((-1)^(n-1))/(n(2n; n))=2/5sqrt(5)csch^(-1)(2)=0.4304089409...
(18)
sum_(n=1)^(infty)((-1)^(n-1))/(n^2(2n; n))=2[csch^(-1)(2)]^2=0.4631296411...
(19)
sum_(n=1)^(infty)((-1)^(n-1))/(n^3(2n; n))=2/5zeta(3)=0.4808227612...
(20)

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

 sum_(n=1)^infty((-1)^(n-1))/(n^k(2n; n))=1/2_(k+1)F_k(1,...,1_()_(k+1);3/2,2,...,2_()_(k-1);-1/4).
(21)

See also

Binomial Coefficient, Binomial Sums, Catalan Number, Central Fibonomial Coefficient, Central Trinomial Coefficient, Erdős Squarefree Conjecture, Staircase Walk, Sárközy's Theorem, Quota System

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References

Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. Experimental Mathematics in Action. Wellesley, MA: A K Peters, 2007.Boros, G. and Moll, V. Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals. Cambridge, England: Cambridge University Press, p. 14, 2004.Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, 1974.Erdős, P.; Graham, R. L.; Ruzsa, I. Z.; and Straus, E. G. "On the Prime Factors of (2n; n)." Math. Comput. 29, 83-92, 1975.Granville, A. and Ramare, O. "Explicit Bounds on Exponential Sums and the Scarcity of Squarefree Binomial Coefficients." Mathematika 43, 73-107, 1996.Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, 1994.Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983.Lehmer, D. H. "Interesting Series Involving the Central Binomial Coefficient." Amer. Math. Monthly 92, 449-457, 1985.Moll, V. H. "Some Questions in the Evaluation of Definite Integrals." MAA Short Course, San Antonio, TX. Jan. 2006. http://crd.lbl.gov/~dhbailey/expmath/maa-course/Moll-MAA.pdf.Plouffe, S. "The Art of Inspired Guessing." Aug. 7, 1998. http://www.lacim.uqam.ca/~plouffe/inspired.html.Sander, J. W. "On Prime Divisors of Binomial Coefficients." Bull. London Math. Soc. 24, 140-142, 1992.Sárkőzy, A. "On Divisors of Binomial Coefficients. I." J. Number Th. 20, 70-80, 1985.Sloane, N. J. A. Sequences A000984/M1645, A001405/M0769, A005836/M2353, A046098, A073010, A073016, A086463, A086464 A086465, A086466, A086467, A086468, A114493, and A114501 in "The On-Line Encyclopedia of Integer Sequences."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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Central Binomial Coefficient

Cite this as:

Weisstein, Eric W. "Central Binomial Coefficient." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CentralBinomialCoefficient.html

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