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Three new classes of binomial Fibonacci sums [PDF]
Transactions on CombinatoricsIn this paper, we introduce three new classes of binomial sums involving Fibonacci (Lucas) numbers and weighted binomial coefficients. One particular result is linked to a problem proposal recently published in the journal The Fibonacci Quarterly.
Robert Frontczak
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Advanced Science, EarlyView.Using a minimal set of experimentally labeled antimicrobial and non‐antimicrobial peptides, we reconstruct the effective lipid composition of a pathogenic membrane ‐demonstrated at the example of bacterial membranes‐ through a physics‐based evolutionary molecular dynamics (Evo‐MD) approach. Optimization based on peptide insertion reveals broad‐spectrum
Maximilian Krebs, Herre Jelger Risselada
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A Note on Extended Binomial Coefficients
J. Integer Seq., 2014 We study the distribution of the extended binomial coefficients by deriving a complete asymptotic expansion with uniform error terms. We obtain the expansion from a local central limit theorem and we state all coefficients explicitly as sums of Hermite polynomials and Bernoulli numbers.
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Applications of Lehmer’s Infinite Series Involving Reciprocals of the Central Binomial Coefficients
Journal of Function Spaces, 2022 The main objective of this paper is to establish several new closed-form evaluations of the generalized hypergeometric function Fq+1qz for q=2,3,4,5. This is achieved by means of separating the generalized hypergeometric function Fq+1qz (q=2,3,4,5) into ...
B. R. Srivatsa Kumar +2 more
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Artificial Intelligence‐Guided Phenotypic Drug Repurposing Against Streptococcus pneumoniae
Advanced Science, EarlyView.Integrated artificial intelligence (AI) ensembles identify highly potent repurposed drugs against Streptococcus pneumoniae. Prospective virtual screening of 6747 drugs uncovers nine active candidates, including thiostrepton. This drug exhibits low picomolar potency and bypasses established resistance in multidrug‐resistant strains, demonstrating an ...
Joshua S. Fitch +6 more
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Sum of the Reciprocals of the Binomial Coefficients
European Journal of Combinatorics, 1993 Let \(S_ n=\sum^ n_{k=0}{1 \over {n \choose k}}\). It is shown that \(S_ n\) satisfies the recurrence \(S_ n={n+1 \over 2n}S_{n-1}+1\). The proof can be simplified by observing that \[ n!S_ n=\sum^ n_{k=0}k!(n-k)!=n!+\sum^{n-1}_{k=0}k!(n-k-1)!(n+1-k-1) =n!+(n+ 1)(n-1)!S_{n-1}-(n!S_ n-n!). \]
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Some divisibility properties of binomial and q -binomial coefficients
Journal of Number Theory, 2014 We first prove that if $a$ has a prime factor not dividing $b$ then there are infinitely many positive integers $n$ such that $\binom {an+bn} {an}$ is not divisible by $bn+1$. This confirms a recent conjecture of Z.-W. Sun. Moreover, we provide some new divisibility properties of binomial coefficients: for example, we prove that $\binom {12n} {3n}$ and
Krattenthaler, Christian +1 more
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Advanced Science, EarlyView.TSTScope is an interpretable AI framework that integrates single‐cell transcriptomes with TCR information through curated gene‐program constraints. By linking receptor context to functional T cell states, it reveals response‐associated tumor‐specific T cell programs in lung cancer immunotherapy cohorts and defines an MPR score associated with ...
Shiwei Cao +8 more
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Some combinatorial identities concerning harmonic numbers and binomial coefficients [PDF]
Discrete Mathematics Letters, 2021 Dongwei Guo
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Note on the binomial partial difference equation
Electronic Journal of Qualitative Theory of Differential Equations, 2015 Some formulas for the "general solution" to the binomial partial difference equation $$c_{m,n}=c_{m-1,n}+c_{m-1,n-1},$$ are known in the literature. However, it seems that there is no such a formula on the most natural domain connected to the equation ...
Stevo Stevic
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