An !-sequence is defined by an = !an−1 − an−2, with initial conditions a0 = 0, a1 = 1. These !-sequences play a remarkable role in partition theory, allowing !generalizations of the Lecture Hall Theorem and Euler’s Partition Theorem. These special properties are not shared with other sequences, such as the Fibonacci sequence, defined by second-order linear recurrences. The !-sequence gives rise...

Diaconis, P. and R. Graham, Binomial coefficient codes over GF(2) Discrete Mathematics 106/107 (1992) 181-188. In this note we study codes over GF(2) which are generated for given d and r by binary vectors of the form ((y), (,‘), , ({), . , (*‘i ‘)) (mod 2), 0 <i =Z d. We describe the weight enumerators of these codes and the numbers of codewords of weights 1 and 2. These results can be used to...

Denote by x a random innnite path in the graph of Pascal's triangle (left and right turns are selected independently with xed probabilities) and by d n (x) the binomial coeecient at the n'th level along the path x. Then for a dense G set of in the unit interval, fd n (x)g is almost surely dense but not uniformly distributed modulo 1.

Denote by x a random infinite path in the graph of Pascal’s triangle (left and right turns are selected independently with fixed probabilities) and by dn(x) the binomial coefficient at the n’th level along the path x. Then for a dense Gδ set of θ in the unit interval, {dn(x)θ} is almost surely dense but not uniformly distributed modulo 1.

In this work, we present a novel method for approximating a normal distribution with a weighted sum of normal distributions. The approximation is used for splitting normally distributed components in a Gaussian mixture filter, such that components have smaller covariances and cause smaller linearization errors when nonlinear measurements are used for the state update. Our splitting method uses ...

In this paper, we present some identities of Gaussian binomial coefficients with respect to recursive sequences, Fibonomial coefficients, and complete functions by use their relationships.

We define an overpartition analogue of Gaussian polynomials (also known as q-binomial coefficients) as a generating function for the number of overpartitions fitting inside the M ×N rectangle. We call these new polynomials over Gaussian polynomials or over q-binomial coefficients. We investigate basic properties and applications of over q-binomial coefficients. In particular, via the recurrence...