Tangent Number

A tangent number is an integer defined for positive integers by

(1)

where is a Bernoulli number. The tangent numbers, also called zag numbers, can also be defined by the Maclaurin series of or as the numbers of alternating permutations on symbols, allowing either alternation pattern, with permutations that are 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 1 and 3 symbols are , and , , respectively.

The tangent numbers have the generating function

			(2)
			(3)
			(4)

for .

Shanks (1967, p. 689) defines generalized tangent numbers for integers and by

(5)

where the Dirichlet L-series is written in Shanks's parameter convention as

(6)

for nonzero integers and complex numbers with , where denotes the Jacobi symbol. The case has a separate normalization,

(7)

The following table gives the first few values of for , 2, .... In the two OEIS records with initial index 0, OEIS A000464 and OEIS A000191, corresponds to the term with index .

	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, ...