Let E be a real quadratic field with discriminant d ≡ 0 (mod p) where p is an odd prime. In terms of a Lucas quotient, the fundamental unit and the class number of E, we determine 0<c<d, (d c)=ρ p−1 ⌊pc/d⌋ modulo p 2 where ρ = ±1.

The binomial coefficients and Catalan triangle numbers appear as weight multiplicities of the finite-dimensional simple Lie algebras and affine Kac–Moody algebras. We prove that any binomial coefficient can be written as weighted sums along rows of the Catalan triangle. The coefficients in the sums form a triangular array, which we call the alternating Jacobsthal triangle. We study various subs...

We prove a general theorem about the multiplicity of the entries in certain integer arrays which is best possible in general. As an application we give non-trivial bounds for the multiplicities of several well-known combinatorial arrays including the binomial coefficients, Narayana numbers and the Eulerian numbers. For the binomial coefficients we obtain the result of Singmaster.

In this paper, we study congruences on sums of products of binomial coefficients that can be proved by using properties of the Jacobi polynomials. We give special attention to polynomial congruences containing Catalan numbers, second-order Catalan numbers, the sequence Sn = ( 3n)( 3n 2n) 2( n )(2n+1) , and the binomial coefficients ( 3n n )

We show how lattice paths and the reflection principle can be used to give easy proofs of unimodality results. In particular, we give a “one-line” combinatorial proof of the unimodality of the binomial coefficients. Other examples include products of binomial coefficients, polynomials related to the Legendre polynomials, and a result connected to a conjecture of Simion.