Tangent numbers: tan(x) = Σ T_n·x^{2n+1}/(2n+1)!. T_n: 1, 2, 16, 272, 7936, 353792... Also: T_n = E_{2n+1} (odd zigzag numbers). Bernoulli relation: T_n = |B_{2n+2}|·2^{2n+2}(2^{2n+2}-1)/(2n+2).
How to Use
Enter n:
- T_n: Tangent number
- Bernoulli: B_{2n+2} link
- Sequence: First values
Bernoulli Connection
T_n = (-1)^n · 2^{2n+2}(2^{2n+2}-1)/(2n+2) · B_{2n+2}. Since Bernoulli numbers involve ζ(2n): T_n = 2^{2n+2}(2^{2n+2}-1)/(2n+2)! · (2n+1)! · |B_{2n+2}|. Deep connection to Riemann zeta.
Combinatorics
T_n counts alternating permutations of {1,...,2n+1} (odd length). Example: T_1=2 counts the 2 alternating permutations of {1,2,3}: 132 and 231. These are the 'down-up' permutations.
Step-by-Step Instructions
- 1Enter n.
- 2Compute T_n.
- 3See Bernoulli.
- 4View sequence.
- 5Verify zigzag.