Piano man. Here is a link in http://www.voofie.com/concept/Mathematics/" [Broken] that you maybe interested.
http://www.voofie.com/content/117/an-explicit-formula-for-the-euler-zigzag-numbers-updown-numbers-from-power-series/" [Broken]
I derived the power series of the function sec x +...
Actually, you can find a close form for your summation expression. Please refer to http://www.voofie.com/concept/Mathematics/" [Broken] for details:
http://www.voofie.com/content/156/how-to-sum-sinn-q-and-cosn-q/" [Broken]
In short,
\sum _{n=-N}^N \cos (n \theta )=\cos (N \theta )+\cot...
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
@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...
I have derived an explicit formula for the http://mathworld.wolfram.com/EulerZigzagNumber.html" [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...
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...
The answer is:
\left(
\begin{array}{c}
n+d+1 \\
n+1
\end{array}
\right)
Please refer to:
http://www.voofie.com/content/76/evaluating-summation-involving-binomial-coefficients/" [Broken]
for the steps and how to deal with problem of this type.