Products and Sums Divisible by Central Binomial Coefficients

In this paper we study products and sums divisible by central binomial coefficients. We show that 2(2n+ 1) ( 2n n ) ∣∣∣∣ (6n 3n )( 3n n ) for all n = 1, 2, 3, . . . . Also, for any nonnegative integers k and n we have ( 2k k ) ∣∣∣∣ (4n+ 2k + 2 2n+ k + 1 )( 2n+ k + 1 2k )( 2n− k + 1 n ) and ( 2k k ) ∣∣∣∣ (2n+ 1)(2n n ) Cn+k ( n+ k + 1 2k ) , where Cm denotes the Catalan number 1 m+1 ( 2m m ) = ( 2m m ) − ( 2m m+1 ) . On the basis of these results, we obtain certain sums divisible by central binomial coefficients.