The $p$-adic valuation of $k$-central binomial coefficients

@article{Straub2009TheV,
  title={The \$p\$-adic valuation of \$k\$-central binomial coefficients},
  author={Armin Straub and Tewodros Amdeberhan and Victor H. Moll},
  journal={Acta Arithmetica},
  year={2009},
  volume={140},
  pages={31-42}
}
The coefficients c(n,k) defined by (1-k^2x)^(-1/k) = sum c(n,k) x^n reduce to the central binomial coefficients for k=2. Motivated by a question of H. Montgomery and H. Shapiro for the case k=3, we prove that c(n,k) are integers and study their divisibility properties. 
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References

SHOWING 1-10 OF 14 REFERENCES
Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients
The distribution of squarefree binomial coefficients . For many years, Paul Erdős has asked intriguing questions concerning the prime divisors of binomial coefficients, and the powers to which they
Introduction to p-adic analytic number theory
This book is an elementary introduction to $p$-adic analysis from the number theory perspective. With over 100 exercises, it will acquaint the non-expert with the basic ideas of the theory and
p -adic Numbers
Having built our foundation, we can now apply the general theory to the specific case of the field ℚ of rational numbers. Extending our scope to include all fields of algebraic numbers (i.e., finite
On the Prime Factors of ( 2 „ " )
Several quantitative results are given expressing the fact that ( ) is usually divisible by a high power of the small primes. On the other hand, it is shown that for any two primes p and q, there
Symmetric Quantum Calculus
The q- and h-differentials may be “symmetrized“ in the following way, $$ \tilde d_q f(x) = f(qx) - f(q^{ - 1} x), $$ (26.1) $$ \tilde d_h g(x) = g(x + h) - g(x - h), $$ (26.2)
On Generalizations of the Stirling Number Triangles 1
Sequences of generalized Stirling numbers of both kinds are introduced. These sequences of triangles (i.e. infinite-dimensional lower triangular matrices) of numbers will be denoted by S2(k;n,m) and
Comp
  • Comp
  • 1975
Théorie des nombres
Theorie des Nombres. Firmin Didot Freres
  • Theorie des Nombres. Firmin Didot Freres
  • 1830
...
1
2
...