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In this paper we introduce and investigate moment generating Stirling numbers of the first kind, "`MSN1"'. They are inverses of MSN2's, which make the representation of the moments for a lot of statistical distributions in closed formulas…
Natural language processing has greatly benefited from the introduction of the attention mechanism. However, standard attention models are of limited interpretability for tasks that involve a series of inference steps. We describe an…
A sequence is called $C$-finite if it satisfies a linear recurrence with constant coefficients. We study sequences which satisfy a linear recurrence with $C$-finite coefficients. Recently, it was shown that such $C^2$-finite sequences…
The paperfolding sequences form an uncountable class of infinite sequences over the alphabet $\{ -1, 1 \}$ that describe the sequence of folds arising from iterated folding of a piece of paper, followed by unfolding. In this note we observe…
Sequence-based modeling broadly refers to algorithms that act on data that is represented as an ordered set of input elements. In particular, Machine Learning algorithms with sequences as inputs have seen successfull applications to…
We prove a recursive identity involving formal iterated logarithms and formal iterated exponentials. These iterated logarithms and exponentials appear in a natural extension of the logarithmic formal calculus used in the study of…
Sequential activation of neurons is a common feature of network activity during a variety of behaviors, including working memory and decision making. Previous network models for sequences and memory emphasized specialized architectures in…
We consider the avoidance of patterns in inversion sequences that relate sorting via sorting machines including data structures such as pop stacks and stacks. Such machines have been studied under a variety of additional constraints and…
Neurons encode and transmit information in spike sequences. However, despite the effort devoted to quantify their information content, little progress has been made in this regard. Here we use a nonlinear method of time-series analysis…
This note examines linear combinations of multi-indexed sequences and derives the multivariate generating function of such a linear combination in terms of the original sequence's m.g.f. Applications include finding distributions and…
Odlyzko and Stanley introduced a greedy algorithm for constructing infinite sequences with no 3-term arithmetic progressions when beginning with a finite set with no 3-term arithmetic progressions. The sequences constructed from this…
By using the matrix formulation of the two-step approach to distributions of patterns in random sequences, recurrence and explicit formulas for the generating functions of successions in random permutations of arbitrary multisets are…
Sequence rotation consists of a circular shift of the sequence's elements by a given number of positions. We present the four classic algorithms to rotate a sequence; the loop invariants underlying their correctness; detailed correctness…
In this paper 101 new integer sequences, sub-sequences, and sequences of sequences, together with related unsolved problems and conjectures, are presented. Also, definitions, examples, solved or open questions, and references for each…
We propose a new efficient iterative method for generating random correlated binary sequences with prescribed correlation function. The method is based on consecutive linear modulations of initially uncorrelated sequence into a correlated…
While many production-ready and robust algorithms are available for the task of recommendation systems, many of these systems do not take the order of user's consumption into account. The order of consumption can be very useful and matters…
Random residue sequences (RR) may be used in many random number applications including those related to multiple access in communications. This paper investigates variations on an algorithm to generate RR sequences that was proposed earlier…
Associated to each complex-valued random variable satisfying appropriate integrability conditions, we introduce a different generalization of the Stirling numbers of the second kind. Various equivalent definitions are provided. Attention,…
We introduce the notion of iterated group extensions, which, roughly speaking, is what one obtains by forming a group extension of a group extension. We interpret iterated extensions in terms of group cohomology, in the same way as…
We define reflective numbers and their iterative summations. We provide classification of reflective numbers based on their iterative cyclical limits.