Euler numbers

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There are (at least) two distinct sequences named Euler numbers, which are related:

The Euler or secant numbers $\scriptstyle E(n)\,$ are defined via the exponential generating function

$G_{\{E(n)\}}(x)\equiv \sum _{n=0}^{\infty }{\frac {E(n)\,x^{n}}{n!}}={\rm {sech}}(x)={\frac {1}{{\rm {cosh}}(x)}},\,$

where $\scriptstyle {\rm {sech}}(x)\,$ is the hyperbolic secant function and $\scriptstyle {\rm {cosh}}(x)\,$ is the hyperbolic cosine function.

A122045 Euler (or secant) numbers $\scriptstyle E(n),\,n\,\geq \,0.\,$

{1, 0, -1, 0, 5, 0, -61, 0, 1385, 0, -50521, 0, 2702765, 0, -199360981, 0, 19391512145, 0, -2404879675441, 0, 370371188237525, 0, -69348874393137901, 0, 15514534163557086905, ...}

A000364 Euler (or secant or "Zig") numbers $\scriptstyle |E(2n)|,\,n\,\geq \,0\,$: $\scriptstyle E(2n)\,$ given by e.g.f. (even powers only) $\scriptstyle {\rm {sech}}(x)\,=\,{\frac {1}{{\rm {cosh}}(x)}}.\,$

{1, 1, 5, 61, 1385, 50521, 2702765, 199360981, 19391512145, 2404879675441, 370371188237525, 69348874393137901, 15514534163557086905, 4087072509293123892361, ...}

The above is a bisection of the following sequence, also called Euler, or up/down numbers:

A000111 Euler or up/down numbers: e.g.f. sec(x) + tan(x). Also for n >= 2, half the number of alternating permutations on n letters (A001250).

{1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521, 353792, 2702765, 22368256, 199360981, 1903757312, 19391512145, 209865342976, 2404879675441, 29088885112832, ...}