Tangent Number

The tangent numbers, also called a zag number, and given by

(1)

where  is a Bernoulli number, are numbers that can be defined either in terms of a generating function given as the Maclaurin series of  or as the numbers of alternating permutations on , 3, 5, 7, ... symbols (where permutations that are the reverses of one another counted as equivalent). The first few  for , 2, ... are 1, 2, 16, 272, 7936, ... (OEIS A000182).

For example, the reversal-nonequivalent alternating permutations on  and 3 numbers are , and , , respectively.

The tangent numbers have the generating function

			(2)
			(3)
			(4)

Shanks (1967) defines a generalization of the tangent numbers by

(5)

where  is a Dirichlet L-series, giving the special case

(6)

The following table gives the first few values of  for , 2, ....

	OEIS
1	A000182	1, 2, 16, 272, 7936, ...
2	A000464	1, 11, 361, 24611, ...
3	A000191	2, 46, 3362, 515086, ...
4	A000318	4, 128, 16384, 4456448, ...
5	A000320	4, 272, 55744, 23750912, ...
6	A000411	6, 522, 152166, 93241002, ...
7	A064072	8, 904, 355688, 296327464, ...
8	A064073	8, 1408, 739328, 806453248, ...
9	A064074	12, 2160, 1415232, 1951153920, ...
10	A064075	14, 3154, 2529614, 4300685074, ...

REFERENCES:

Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.

Knuth, D. E. and Buckholtz, T. J. "Computation of Tangent, Euler, and Bernoulli Numbers." Math. Comput. 21, 663-688, 1967.

Shanks, D. "Generalized Euler and Class Numbers." Math. Comput. 21, 689-694, 1967.

Shanks, D. Corrigendum to "Generalized Euler and Class Numbers." Math. Comput. 22, 699, 1968.

Sloane, N. J. A. Sequence A000182/M2096 in "The On-Line Encyclopedia of Integer Sequences."