The tangent numbers, also called a zag number, and given by
(1)
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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)
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(3)
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(4)
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Shanks (1967) defines a generalization of the tangent numbers by
(5)
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where is a Dirichlet L-series, giving the special case
(6)
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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, ... |