Formulas for powers of the hyperbolic tangent with an application to higher-order tangent numbers

It is shown that the function tanh(2n+1)(x) is a linear combination of even-order derivatives of tanh(x), while the function 1 - tanh(2n+2)(x) is a linear combination of odd-order derivatives of tanh(x). These results are then used to express higher-order tangent numbers (coefficients in the Maclaurin series for tanh(n)(x)) as linear combinations of the ordinary tangent numbers (coefficients in the Maclaurin series for tanh(x)).