We present six identities involving Stirling numbers of both kinds. The first two convolution identities appear to be new, and we prove them by algebraic methods using falling powers and generating functions. The other four are recurrence relations for Stirling numbers that recently appeared as conjectures. Two of these were proved, in equivalent forms, elsewhere by algebraic methods. However, we provide novel combinatorial proofs for all of them.
The OEIS sequence A080416, proposed by Barry, is a Stirling-like number triangle defined by a paired decomposition of C(n + 3, 3). The recurrence formula for this sequence can be derived from an article by Corcino et al. However, explicit formulas have not been discovered. In this paper, we derive two formulas for calculating the members of this sequence. One of them involves matrix multiplication, while the other one does not require it. We also generalize the approach for sequences defined by counting North-East lattice paths with weights.
For p, q E N, a finite nonempty set F is said to be (p, q)-Schreier (or maximal (p, q)Schreier, respectively) if q min F >= p F (or q min F = p F , respectively). Using the of (p, q)-Schreier sets of the natural numbers. We show that the counts are taken periodically from Padovan-like sequences that satisfy simple recurrence relations. As an application, we obtain an alternative proof of Beanland et al.'s result. Furthermore, a similar result holds for the counts of maximal (p, q)-Schreier sets. We end with a discussion of the relation between (p, q)-Schreier and maximal (p, q)-Schreier sets.
We find formulas for the determinants of several Toeplitz-Hessenberg matrices whose nonzero entries are Fuss-Catalan numbers. By Trudi's formula, one obtains equivalent multi-sum identities indexed over the set of partitions of a fixed integer involving the product of Fuss-Catalan numbers and multinomial coefficients. We make use of generating functions to establish our results and, as a consequence, several entries from the OEIS are afforded new combinatorial interpretations as determinants of certain Toeplitz-Hessenberg matrices. Finally, we provide counting arguments for several of our determinant identities drawing upon combinatorial interpretations of various specific cases of the Fuss-Catalan numbers in terms of Dyck paths, ternary trees, and non-crossing partitions.
A pseudoprime is a composite integer N that mimics the behavior of primes by satisfying the congruence 2N equivalent to 2 (mod N) in Fermat's little theorem. This paper focuses on the subset of even pseudoprimes and obtains an upper bound for the sum of their reciprocals. Our approach combines analytic arguments with computational verification, showing that this sum is less than 0.0059.
