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  1. R

    Deriving tan power series

    Piano man. Here is a link in" [Broken] that you maybe interested." [Broken] I derived the power series of the function sec x +...
  2. R

    Another infinite series

    Actually, you can find a close form for your summation expression. Please refer to" [Broken] for details:" [Broken] In short, \sum _{n=-N}^N \cos (n \theta )=\cos (N \theta )+\cot...
  3. R

    AMC Question HARD

    You may just write out the recurrence equation and solve it yourself: a_n = \frac{a_{n+1}+a_{n-1}}{2} +1 a_1=1 a_{12}=12 The solution is: a_n = -n^2 + 14 n -12
  4. R

    Explicit formula for Euler zigzag numbers(Up/down numbers)

    @arildno Thank you for the link. I haven't read the "Entringer numbers" before. But I don't found explicit formula, and seems the Euler zigzag number is it's special case. @CRGreathouse I have read the link you send me before. Thank you. First, I found out the formula, not because it is...
  5. R

    Explicit formula for Euler zigzag numbers(Up/down numbers)

    I have derived an explicit formula for the" [Broken], the number of alternating permutations for n elements: A_j=i^{j+1}\sum _{n=1}^{j+1} \sum _{k=0}^n \frac{C_k^n(n-2k)^{j+1}(-1)^k}{2^ni^nn} For details, please refer to my article in...
  6. R

    Equation of a circle

    In short, you are dealing with the case where R > A. In this case, value of \theta cannot be arbitrary. If the value of \theta is too large, there will be no intersection between the radial line and the circle. Please refer to this...
  7. R

    Summation combinatorics

    The answer is: \left( \begin{array}{c} n+d+1 \\ n+1 \end{array} \right) Please refer to:" [Broken] for the steps and how to deal with problem of this type.