Results 11 to 20 of about 165,496 (329)
A Fast Algorithm for Computing Binomial Coefficients Modulo Powers of Two [PDF]
The Scientific World Journal, 2013 I present a new algorithm for computing binomial coefficients modulo . The proposed method has an preprocessing time, after which a binomial coefficient with can be computed modulo in time.
Mugurel Ionut Andreica
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Generating binomial coefficients in a row of Pascal's triangle from extensions of powers of eleven [PDF]
Heliyon, 2022 Sir Isaac Newton noticed that the values of the first five rows of Pascal's triangle are each formed by a power of 11, and claimed that subsequent rows can also be generated by a power of 11.
Md. Shariful Islam +3 more
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Bisecting binomial coefficients [PDF]
Discrete Applied Mathematics, 2016 In this paper, we deal with the problem of bisecting binomial coefficients. We find many (previously unknown) infinite classes of integers which admit nontrivial bisections, and a class with only trivial bisections.
Ionaşcu, Eugen J. +2 more
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On alternating sums of binomial coefficients and $q$-binomial coefficients [PDF]
, 2016 In this paper we shall evaluate two alternating sums of binomial coefficients by a combinatorial argument. Moreover, by combining the same combinatorial idea with partition theoretic techniques, we provide $q$-analogues involving the $q$-binomial coefficients.
Mohamed El Bachraoui
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Sums involving the binomial coefficients, Bernoulli numbers of the second kind and harmonic numbers [PDF]
Notes on Number Theory and Discrete Mathematics, 2023 We offer a number of various finite and infinite sum identities involving the binomial coefficients, Bernoulli numbers of the second kind and harmonic numbers. For example, among many others, we prove Σⁿₖ₌ₒ((-1)ᵏhₖ/4ᵏ)$binom{2k}{k}$Gₙ₋ₖ = ((-1)ⁿ⁻¹/2^²ⁿ⁻¹)
Necdet Batır, Anthony Sofo
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Supercongruences involving Apéry-like numbers and binomial coefficients
AIMS Mathematics, 2022 Let $ \{S_n\} $ be the Apéry-like sequence given by $ S_n = \sum_{k = 0}^n\binom nk\binom{2k}k\binom{2n-2k}{n-k} $. We show that for any odd prime $ p $, $ \sum_{n = 1}^{p-1}\frac {nS_n}{8^n}{\equiv} (1-(-1)^{\frac{p-1}2})p^2\ (\text{ mod}\ {p^3}) $. Let
Zhi-Hong Sun
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Dirichlet series and series with Stirling numbers
Cubo, 2023 This paper presents a number of identities for Dirichlet series and series with Stirling numbers of the first kind. As coefficients for the Dirichlet series we use Cauchy numbers of the first and second kinds, hyperharmonic numbers, derangement numbers ...
Khristo Boyadzhiev
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Convolution identities involving the central binomial coefficients and Catalan numbers [PDF]
Transactions on Combinatorics, 2021 We generalize some convolution identities due to Witula and Qi et al. involving the central binomial coefficients and Catalan numbers. Our formula allows us to establish many new identities involving these important quantities, and recovers some ...
Necdet Batır, Hakan Kucuk, Sezer Sorgun
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A class of symmetric and non-symmetric band matrices via binomial coefficients
Special Matrices, 2021 Symmetric matrix classes of bandwidth 2r + 1 was studied in 1972 through binomial coefficients. In this paper, non-symmetric matrix classes with the binomial coefficients are considered where r + s + 1 is the bandwidth, r is the lower bandwidth and s is ...
Micheal Omojola, Kilic Emrah
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Generalized double Fibonomial numbers
Ratio Mathematica, 2021 From the beginning of 20th century, generalization of binomial coefficient has been deliberated broadly. One of the most famous generalized binomial coefficients are Fibonomial coefficients, obtained by substituting Fibonacci numbers in place of natural ...
Mansi Shah, Shah Devbhadra
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