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Journal of Number Theory, ISSN 0022-314X, 02/2020, Volume 207, pp. 145 - 155
A practical number is a positive integer n such that every positive integer less than n can be written as a sum of distinct divisors of n. We prove that most... 
Binomial coefficient | Central binomial coefficient | Practical number | MATHEMATICS | DIVISORS
Journal Article
Advances in Applied Mathematics, ISSN 0196-8858, 2010, Volume 45, Issue 1, pp. 125 - 148
Let p be a prime and let a be a positive integer. In this paper we determine ∑ k = 0 p a − 1 ( 2 k k + d ) / m k and ∑ k = 1 p − 1 ( 2 k k + d ) / ( k m k − 1... 
Central binomial coefficients | Bernoulli numbers | Fibonacci numbers | Congruences modulo primes | MATHEMATICS, APPLIED | NUMBERS | WIEFERICH | THEOREM | SEARCH | WILSON | FERMAT | PRIMES | Integers | Congruences | Binomial coefficients | Mathematical analysis | Symbols
Journal Article
Journal of Number Theory, ISSN 0022-314X, 2011, Volume 131, Issue 11, pp. 2219 - 2238
It is known that ∑ k = 0 ∞ ( 2 k k ) ( 2 k + 1 ) 4 k = π 2 and ∑ k = 0 ∞ ( 2 k k ) ( 2 k + 1 ) 16 k = π 3 . In this paper we obtain their p-adic analogues such... 
Central binomial coefficients | Congruences modulo prime powers | Euler numbers | Binary quadratic forms | Secondary | Primary | MATHEMATICS | BERNOULLI NUMBERS
Journal Article
The Ramanujan Journal, ISSN 1382-4090, 6/2019, Volume 49, Issue 2, pp. 237 - 256
Let $$p>3$$ p > 3 be a prime and let a be a positive integer. We show that if or $$a>1$$ a > 1 , then with $$(-)$$ ( - ) the Jacobi symbol, which confirms a... 
11A07 | Congruences | Functions of a Complex Variable | Secondary 05A10 | Field Theory and Polynomials | Primary 11B65 | Mathematics | 11B68 | Legendre symbol | Fourier Analysis | Central binomial coefficients | Number Theory | Combinatorics | MATHEMATICS
Journal Article
Finite Fields and Their Applications, ISSN 1071-5797, 07/2013, Volume 22, pp. 24 - 44
In this paper we deduce some new supercongruences modulo powers of a prime p>3. Let d∈{0,1,…,(p−1)/2}. We show... 
Central binomial coefficients | Supercongruences modulo prime powers | MATHEMATICS | MATHEMATICS, APPLIED | ANALOGS | CONGRUENCES
Journal Article
Journal of Number Theory, ISSN 0022-314X, 11/2018, Volume 192, pp. 221 - 239
For nonnegative integers j and n let Θ(j,n) be the number of entries in the n-th row of Pascal's triangle that are not divisible by 2j+1. In this paper we... 
Central limit law | Binomial coefficients | Multivariate asymptotics of generating functions | Divisibility by powers of primes | MATHEMATICS | PRIME | NUMBER | DIVIDES
Journal Article
American Mathematical Monthly, ISSN 0002-9890, 04/2014, Volume 121, Issue 4, pp. 344 - 349
We derive asymptotic formulas for central extended binomial coefficients, which are generalizations of binomial coefficients, using the distribution of the sum... 
MATHEMATICS | THEOREM | Usage | Approximation theory | Central limit theorem | Asymptotes | Binomial coefficients | Analysis
Journal Article
International Journal of Number Theory, ISSN 1793-0421, 05/2018, Volume 14, Issue 4, pp. 1135 - 1141
Let be the set of all positive integers n such that n divides the central binomial coefficient 2 n n . Pomerance proved that the upper density of is at most 1... 
upper and lower densities | Central binomial coefficient | divisibility | MATHEMATICS | NUMBERS
Journal Article
Colloquium Mathematicum, ISSN 0010-1354, 2015, Volume 139, Issue 1, pp. 127 - 136
Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let p > 3 be a prime. We show that Tp-1 equivalent... 
Central trinomial coefficients | Lucas sequences | Congruences | Binomial coefficients | MATHEMATICS | congruences | binomial coefficients | central trinomial coefficients
Journal Article
Advances in Applied Mathematics, ISSN 0196-8858, 2010, Volume 45, Issue 3, pp. 303 - 316
Motivated by recent works of Sun and Tauraso, we prove some variations on the Green–Krammer identity involving central q-binomial coefficients, such as ∑ k = 0... 
q-Binomial coefficient | Cyclotomic polynomial | Central binomial coefficients | Congruence | MATHEMATICS, APPLIED | THEOREM | Q-FIBONACCI POLYNOMIALS | Employee motivation | Congruences | Mathematical analysis | Sun | Symbols
Journal Article
Tatra Mountains Mathematical Publications, ISSN 1210-3195, 09/2017, Volume 70, Issue 1, pp. 199 - 206
In this note we compute the generating function for the numbers terms of elementary functions and dilogarithms. 
Euler series transformation | binomial identities | central binomial coefficients | reciprocals of binomial coefficients | generating functions | harmonic numbers | 05A15 | 11B05
Journal Article
Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 04/2017, Volume 448, Issue 2, pp. 1061 - 1078
In this paper, we confirm several conjectured congruences of Sun concerning the divisibility of binomial sums. For example, with help of a quadratic... 
Congruence | Pell number | Central binomial coefficient | LEGENDRE POLYNOMIALS | MATHEMATICS | MATHEMATICS, APPLIED | RODRIGUEZ-VILLEGAS | ANALOGS | HYPERGEOMETRIC-SERIES | CONGRUENCES | CONJECTURE
Journal Article
The Ramanujan Journal, ISSN 1382-4090, 2/2018, Volume 45, Issue 2, pp. 319 - 330
In this paper, we prove some congruences conjectured by Z.-W. Sun: For any prime $$p>3$$ p > 3 , we determine $$\begin{aligned} \sum \limits _{k = 0}^{p - 1}... 
11A07 | Congruences | Functions of a Complex Variable | Secondary 05A10 | Field Theory and Polynomials | Primary 11B65 | Mathematics | 11B68 | Delannoy numbers | Fourier Analysis | Central binomial coefficients | Harmonic numbers | Number Theory | Catalan numbers | Combinatorics
Journal Article
by Mao, GS and Zhang, T
RAMANUJAN JOURNAL, ISSN 1382-4090, 10/2019, Volume 50, Issue 1, pp. 1 - 11
In this paper, we prove two conjectures of Z.-W. Sun: 2n((2n)(n))vertical bar Sigma(n-1)(k=0)(3k + 1)((2k)(k))(3)16(n-1-k) for all n = 2, 3, ..., and... 
MATHEMATICS | Divisibility problem | Central binomial coefficients | Congruences
Journal Article
Electronic Journal of Combinatorics, ISSN 1077-8926, 01/2013, Volume 20, Issue 1
In this paper we study products and sums divisible by central binomial coefficients. We show that $$2(2n+1)\binom{2n}n\ \bigg|\ \binom{6n}{3n}\binom{3n}n\ \... 
Divisibility | Central binomial coefficients | Congruences
Journal Article
Electronic Journal of Combinatorics, ISSN 1077-8926, 02/2014, Volume 21, Issue 1
Recently, Z. Sun proved that [GRAPHICS] for m is an element of Z(>0). In this paper, we consider a generalization of this result by defining b(n,k) = 2k (n +... 
Coefficients | Central | Binomial | MATHEMATICS | MATHEMATICS, APPLIED | PRODUCTS | CONGRUENCES | central binomial coefficients
Journal Article
Bulletin of the Korean Mathematical Society, ISSN 1015-8634, 2017, Volume 54, Issue 1, pp. 225 - 242
A family of Apery-like series involving reciprocals of central binomial coefficients is studied and it is shown that they represent transcendental numbers. The... 
Transcendental number | Central binomial coef- ficient | Competitive cheap talk | Apéry-like series | Monotonicity | MATHEMATICS | transcendental number | Apery-like series | competitive cheap talk | RAMANUJAN | central binomial coefficient | monotonicity | SPIRIT
Journal Article
Turkish Journal of Mathematics, ISSN 1300-0098, 2016, Volume 40, Issue 5, pp. 973 - 985
In this paper, using some combinatorial identities, we present new congruences involving central binomial coefficients and harmonic, Catalan, and Fibonacci... 
Central binomial coefficients | Catalan numbers | Fibonacci numbers | Harmonic numbers | Pell numbers | MATHEMATICS | P-ADIC CONGRUENCES | harmonic numbers | BERNOULLI
Journal Article
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