Central Binomial Coefficient

Author

Eric W. Weisstein
January 18, 2006

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Combinatorics/CentralBinomialCoefficient.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/CentralBinomialCoefficient.html.

©2006 Wolfram Research, Inc. except for portions noted otherwise

Normal Definition

Table[CentralBinomial[n],{n,10}]

Table[Binomial[2n,n],{n,10}]

Table[(((2^n) (-1+2 n)!!)/(n!)),{n,10}]

Table[(((2n)!)/(((n!))^2)),{n,10}]

Powers of 10

Limit

FullSimplify[Limit[Log[10,CentralBinomial[n]]/n,n->∞]]

N[%,10]

Enumeration

Table[Digits[CentralBinomial[10^n]],{n,0,6}]//Timing

Table[Digits[CentralBinomial[10^n]],{n,0,7}]//Timing

Table[Digits[CentralBinomial[10^n]],{n,0,8}]//Timing

Series

expr=Simplify[Normal[Series[Log[10,CentralBinomial[x]],{x,∞,1}]/.x->10^n],n∈Integers&&n>0]

Table[1+Floor[expr],{n,0,20}]

Series

Series[1/Sqrt[1-4x],{x,0,10}]

<<DiscreteMath`RSolve`

FullSimplify[SeriesTerm[1/Sqrt[1-4x],{x,0,n}],n∈Integers&&n>0]

<<Utilities`Simplify`

FullSimplify[GammaSimplify[%30]]

CentralBinomial[Range[5 10^3]];//Timing

Table[(((2^n) (-1+2 n)!!)/(n!)),{n,5 10^3}];//Timing

Table[(((2n)!)/(((n!))^2)),{n,5 10^3}];//Timing

Integral

(1/π)Integrate[(1/((((x^2)+1))^(m+1))),{x,0,∞}]

(1/π)Integrate[(1/((((x^2)+1))^(m+1))),{x,0,∞},Assumptions->Re[m]>-(1/2)]//FullSimplify

Subtract@@{%,(2^(-2m-1))Binomial[2m,m]}//FullSimplify

Plots

<<Utilities`Plot`

ComplexPlot3D[CentralBinomial[z],{z,-3,3},PlotPoints->100]//Timing

ComplexContourPlot[CentralBinomial[z],{z,-3,3}]//Timing

Primes

Position[CentralBinomial[Range[10^3]],_?PrimeQ]//Flatten//Timing

Divisors

(Plus@@Last/@#)&/@FactorInteger/@CentralBinomial[Range[20]]

FactorInteger/@CentralBinomial[Range[20]]//ColumnForm

Congruences

(({{2 p}, {p}}))≡2 (mod (p^3))

Mod[Binomial[2#,#],#^3]&/@Prime[Range[5]]

Mod[Binomial[2#,#],#^3]&/@Prime[Range[3,1000]]//Union

3 Divides (({{2 n}, {n}}))

Flatten[Position[Mod[CentralBinomial[Range[100]],3],_?(#!=0&)]]

Position[DigitCount[#,3,2]&/@Range[100],0]//Flatten

Squarefree

(({{n}, {⌊n/2⌋}}))

Refine[Floor[n/2],Mod[n,2]==1]//Simplify

Position[Binomial[#,Floor[#/2]]&/@Range[10^3],_?SquareFreeQ]//Flatten//Timing

Select[Range[1,10^5,2],SquareFreeQ[Binomial[#,(#-1)/2]]&]//Timing

(({{2n}, {n}}))

Position[Binomial[2#,#]&/@Range[10^3],_?SquareFreeQ]//Flatten//Timing

(p=Position[Binomial[2#,#]&/@Range[10^4],_?SquareFreeQ])//Flatten//Timing

Position[Binomial[2#,#]&/@Range[10^5],_?SquareFreeQ]//Flatten//Timing

Sums

Without PolyGammas

{#,N[#,20]}&@Sum[1/Binomial[2n,n],{n,∞}]

{#,N[#,20]}&@Sum[1/n/Binomial[2n,n],{n,∞}]

{#,N[#,20]}&@Sum[1/n^2/Binomial[2n,n],{n,∞}]

{#,N[#,20]}&@Sum[1/n^4/Binomial[2n,n],{n,∞}]

N[Zeta[4]17/36,20]

With PolyGammas

FullSimplify[Sum[1/n^3/Binomial[2n,n],{n,∞}]]

{NSum[1/n^3/Binomial[2n,n],{n,∞}],N[%]}

Alternating Sums

{#,N[#,20]}&@Sum[(-1)^(n-1)/Binomial[2n,n],{n,∞}]

{#,N[#,20]}&@Sum[(-1)^(n-1)/n/Binomial[2n,n],{n,∞}]

{#,N[#,20]}&@Sum[(-1)^(n-1)/n^2/Binomial[2n,n],{n,∞}]

{#,N[#,20]}&@Sum[(-1)^(n-1)/n^3/Binomial[2n,n],{n,∞}]

Alternate Definition

Series[((1-4(x^2)-(Sqrt[1-4(x^2)]))/(2(2(x^3)-(x^2)))),{x,0,10}]