Euler zigzag numbers: E_0=1, E_1=1, E_2=1, E_3=2, E_4=5, E_5=16, E_6=61, E_7=272... Count alternating permutations of [n]. Generating function: sec(x)+tan(x) = Σ E_n·x^n/n!. Even indices: secant numbers. Odd: tangent numbers.
How to Use
Enter n:
- E_n: Zigzag number
- Sequence: E_0..E_n
- Type: Secant or tangent
Alternating Perms
1: {1}. 2: {1,2}→only 1 (trivially). 3: all down-up perms of {1,2,3}: 132, 231 → E_3=2. For n=4: 2143, 3142, 3241, 4132, 4231 → E_4=5. Each is a zigzag: up-down-up-down...
Analysis Connection
sec(x) = Σ E_{2n}·x^{2n}/(2n)! and tan(x) = Σ E_{2n+1}·x^{2n+1}/(2n+1)!. So zigzag numbers encode the Taylor series of sec and tan simultaneously!
Step-by-Step Instructions
- 1Enter n.
- 2Compute E_n.
- 3Classify sec/tan.
- 4View sequence.
- 5See growth.