Definition:Euler Numbers/Sequence

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Definition

The sequence of Euler numbers begins:

\(\displaystyle E_0\) \(=\) \(\displaystyle 1\)
\(\displaystyle E_2\) \(=\) \(\displaystyle -1\)
\(\displaystyle E_4\) \(=\) \(\displaystyle 5\)
\(\displaystyle E_6\) \(=\) \(\displaystyle -61\)
\(\displaystyle E_8\) \(=\) \(\displaystyle 1385\)
\(\displaystyle E_{10}\) \(=\) \(\displaystyle -50 \, 521\)
\(\displaystyle E_{12}\) \(=\) \(\displaystyle 2 \, 702 \, 765\)
\(\displaystyle E_{14}\) \(=\) \(\displaystyle -199 \, 360 \, 981\)
\(\displaystyle E_{16}\) \(=\) \(\displaystyle 19 \, 391 \, 512 \, 145\)
\(\displaystyle E_{18}\) \(=\) \(\displaystyle -2 \, 404 \, 879 \, 675 \, 441\)
\(\displaystyle E_{20}\) \(=\) \(\displaystyle 370 \, 371 \, 188 \, 237 \, 525\)
\(\displaystyle E_{22}\) \(=\) \(\displaystyle -69 \, 348 \, 874 \, 393 \, 137 \, 901\)
\(\displaystyle E_{24}\) \(=\) \(\displaystyle 15 \, 514 \, 534 \, 163 \, 557 \, 086 \, 905\)

Odd index Euler numbers are all $0$.

This sequence is A122045 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Sources