We show that the super Catalan numbers are special values of the Krawtchouk polynomials by deriving an expression for the super Catalan numbers in terms of a signed set.

In 2008, Satoshi Nakamoto* famously invented bitcoin, and in his (or her, or their, or its) white paper sketched an approximate formula for the probability of a successful double spending attack by a dishonest party. This was corrected by Meni Rosenfeld, who, under more realistic assumptions, gave the exact probability (missing a foundational proof ); and another formula (along with foundational proof ), in terms of the Incomplete Beta function, was given later by Cyril Grunspan and Ricardo Pérez-Marco, that enabled them to derive an asymptotic formula for that quantity.Using Wilf-Zeilberger algorithmic proof theory, we continue in this vein and present a recurrence equation for the above-mentioned probability of success, that enables a very fast compilation of these probabilities. We next use this recurrence to derive (in algorithmic fashion) higher-order asymptotic formulas, extending the formula of Grunspan and Pérez-Marco who only did the leading term. We then study the statistical properties (expectation, variance, etc.) of the duration of a successful attack.where readers can also find numerous input and output files.In due course, the current package will also be available along with an implementation in Math-Cognify's own symbolic language B from https://github.com/MathCognifyTechnologies/fr-crypto-bitcoin-1. A Two-Phase Soccer MatchIn order to make this article self-contained and focus on the core combinatorial structure that underpins much of the process of the double spend attack, we will postpone the malevolent language of cyber-attacks until the penultimate section of this article, i.e., the notes and remarks section. There we provide context and references to the double-spending attack for the interested reader; additionally, we pinpoint weaknesses and inconsistencies in Dr. Satoshi Nakamoto's paper -strengthening our belief that he is (or was) a competent scientist but not a combinatorialist. For now, we provide an equivalent model featuring a two-phase Soccer match, where one of the two teams is worse than the other.

We count the number of walks of length n on a k-node circular digraph that cover all k nodes in two ways. The first way illustrates the transfer-matrix method. The second involves counting various classes of height-restricted lattice paths. We observe that the results also count so-called k-balanced strings of length n, generalizing a 1996 Putnam problem.

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We set a lower bound on the complexity of options pricing formulae in the lattice metric by proving that no general explicit or closed form (hypergeometric) expression for pricing vanilla European call and put options exists when employing the binomial lattice approach. Our proof follows from Gosper's algorithm.