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Voofie | An Explicit Formula for the Euler zigzag nu... https://web.archive.org/web/20120511083735/http:/... home | FAQ | Languages▼ | login | register All An Explicit Formula for the Euler zigzag numbers (Up/down numbers) from power series Page Info 15 Impacts English , Euler number , Mathematics , Advance , 26 Jul 10 Alternating permutation , Academic Paper In this paper, I will derive an explicit formula for the Euler zigzag numbers (Up/down numbers). Euler zigzag number is the number of alternating permutation in a set. Therefore the explicit formula of Euler numbers(Secant numbers) and Bernoulli numbers are found as well. The formula involves two finite sum. 0 /0 rates 3293 Your Rating: Version: 6 Last update: 02 Sep 10 History Permalink Author Ross Tang (ross_tang) Content Degree in Physics and Mathematics, Master 1 Introduction 2 Integrand of sec(x) + tan(x) in Physics 香港 3 Power Series Expansion of f(x) 4 Simplification 5 Explicit Formula for Euler number 6 Explicit Formula for Bernoulli Numbers 7 Conclusion Mathematics Euler number Alternating permutation 809 0 0 Introduction Euler zigzag numbers, , is the number of alternating permutation of the set {1,2,...,n}. And it is well known that: In the following section, I am going to derive an explicit formula of by using power series expansion. Related Integrand of sec(x) + tan(x) Let's consider the integrand of sec(x) + tan(x): Explore exciting communities of 1 of 5 Repeated sum and partial difference equation Euler number , Mathematics , Alternating permutation , 12/28/2013 05:54 PM Voofie | An Explicit Formula for the Euler zigzag nu... https://web.archive.org/web/20120511083735/http:/... e^-lambda/2 e^A e^B if [A,B]=lambda Solving linear non-homogeneous ordinary differential equation with variable coefficients with operator method Solving a partial difference equation in 2 variables with operator method Let: Solving recurrence equation with indexes from negative infinity to positive infinity I will do a power expansion of the function f(x), and a finite sum explicit formula for can be found by some simplification. Reducing a partial difference equation into a partial differential equation and solving for the generating function using method of characteristics Power Series Expansion of f(x) Binomial Expansion for non-commutative elements (A+B)^n where [A, B] = lambda Finding nth derivative of the function sec x + tan x and partial difference equation A procedure to list all derangements of a multiset A free ebook about generating function, "generatingfunctionology" Therefore, by equating coefficients, we have: or Now, I am going to simplify it, and show that it is actually Explore exciting communities of Euler number , Mathematics , Alternating permutation , 2 of 5 12/28/2013 05:54 PM Voofie | An Explicit Formula for the Euler zigzag nu... https://web.archive.org/web/20120511083735/http:/... equals to: Simplification In order to reduce the infinite sum to a finite sum, I first let: So that: I observed that: To show that, let's define the translational operator that: such Then: Here, is the backward difference operator. Now, consider: The result is a polynomial of degree . We can see that, if we apply backward difference operator to a polynomial, its degree decreases by 1. Therefore, if we apply the backward difference operator for a number of time which is larger than the degree of a polynomial, the result is zero. Explore exciting communities of Euler number , Mathematics , Alternating permutation , 3 of 5 12/28/2013 05:54 PM Voofie | An Explicit Formula for the Euler zigzag nu... As a result, we have https://web.archive.org/web/20120511083735/http:/... . Therefore: Explicit Formula for Euler number Euler number is given by the generating function: And it is given by: Explicit Formula for Bernoulli Numbers From Wikipedia, we know: Therefore: Conclusion I have found out an simple formula for the Euler zigzag number: 0 Comments Impact Reply Newest Oldest Please login to post comment. Explore exciting communities of 4 of 5 Euler number , Mathematics , Alternating permutation , 12/28/2013 05:54 PM Voofie | An Explicit Formula for the Euler zigzag nu... https://web.archive.org/web/20120511083735/http:/... What is Voofie? Voofie organizes knowledge, discovers useful resources and recognizes knowledgable users. Bookmark your blog in Voofie to get more traffic as well as building a reputation in your field! Explore more about it. Become a member—our FREE Registration takes just seconds. About Voofie | FAQ | Terms | Privacy © Voofie Limited Explore exciting communities of 5 of 5 Euler number , Mathematics , Alternating permutation , 12/28/2013 05:54 PM