in (\[5.2\]) and, consequently, $\lambda_{\infty}^d$ are integer multiples of $t_r^d$. With that in mind, yet another, somewhat surprising conjecture emerged from the test results: $$\lambda_{\infty}^d = \begin{cases}
    t_r^d \cdot T_d = \frac{1}{d!} T_d, & \hbox{$d$ odd,} \\[6pt]
    t_r^d \cdot 2^d E_d = \frac{1}{d!} E_d, & \hbox{$d$ even,}
  \end{cases}
  \label{5.3}$$ where, as in [@Knuth-Buckholtz-67], $T_n$ are the tangent numbers, and $E_n$ are the Euler numbers, defined by the Taylor expansions of $\tan t$ and $\sec t$, respectively, $$\tan t = \sum_{n = 0}^\infty T_n \frac{t^n}{n!}, \quad
  \sec t = \sum_{n = 0}^\infty E_n \frac{t^n}{n!}.$$ These numbers are also related to the sequences A000182 (the tangent or “zag” numbers), A000364 (the Euler or “zig” numbers) and A002436, from [@Sloane-2008].

If true, (\[5.3\]) would be of significant practical merit, for there exist very stable and elegant algorithms for calculation of $T_n$ and $E_n$ by Knuth and Buckholtz [@Knuth-Buckholtz-67]. So, it deserved an effort to find the proof.

A unifying framework for handling both cases is provided by the Euler polynomials $E_n(x)$, defined by the following exponential generating function (see [@Abramowitz-Stegun-72 23.1.1, p. 804]) $$\frac{2 e^{xt}}{e^t + 1} = \sum_{n = 0}^\infty E_n(x) \frac{t^n}{n!},
  \label{5.4}$$ which is valid for $|t| < \pi$.

First, note 