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Tangent Number

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

T_n=(2^(2n)(2^(2n)-1)|B_(2n)|)/(2n),
(1)

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

For example, the reversal-nonequivalent alternating permutations on n=1 and 3 numbers are {1}, and {1,3,2}, {2,1,3}, respectively.

The tangent numbers have the generating function

tanx=sum_(k=0)^(infty)((-1)^(k-1)2^(2k)(2^(2k)-1)B_(2k))/((2k)!)x^(2k-1)
(2)
=sum_(k=1)^(infty)(T_k)/((2k-1)!)x^(2k-1)
(3)
=x+1/3x^3+2/(15)x^5+(17)/(315)x^7+....
(4)

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

d_(a,n)=((2n-1)!L_(-a)(2n+1))/(sqrt(a))((2a)/pi)^(2n),
(5)

where L_n(s) is a Dirichlet L-series, giving the special case

d_(1,n)=T_n.
(6)

The following table gives the first few values of d_(a,n) for n=1, 2, ....

aOEISd_(a,n)
1A0001821, 2, 16, 272, 7936, ...
2A0004641, 11, 361, 24611, ...
3A0001912, 46, 3362, 515086, ...
4A0003184, 128, 16384, 4456448, ...
5A0003204, 272, 55744, 23750912, ...
6A0004116, 522, 152166, 93241002, ...
7A0640728, 904, 355688, 296327464, ...
8A0640738, 1408, 739328, 806453248, ...
9A06407412, 2160, 1415232, 1951153920, ...
10A06407514, 3154, 2529614, 4300685074, ...

See also

Alternating Permutation, Dirichlet L-Series, Entringer Number, Euler Zigzag Number, Secant Number, Tangent

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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."

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Tangent Number

Cite this as:

Weisstein, Eric W. "Tangent Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TangentNumber.html

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