On Arithmetic and Asymptotic Properties
نویسندگان
چکیده
On arithmetic and asymptotic properties of up-down numbers. We prove that the set of all up-down numbers C(σ) can be expressed by a single universal polynomial Φ, whose coefficients are products of numbers from the Taylor series of the hyperbolic tangent function. We prove that Φ is a modified exponential, and deduce some remarkable congruence properties for the set of all numbers C(σ), for fixed N. We prove a concise upper-bound for C(σ), which describes the as-ymptotic behaviour of the up-down function C(σ) in the limit C(σ) ≪ (N +1)!.
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