Definition:Euler Numbers/Alternative Form/Sequence

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Definition

The sequence of the alternative form of Euler numbers begins:

\(\ds E_1^*\) \(=\) \(\ds 1\) \(\ds = -E_2\)
\(\ds E_2^*\) \(=\) \(\ds 5\) \(\ds = E_4\)
\(\ds E_3^*\) \(=\) \(\ds 61\) \(\ds = -E_6\)
\(\ds E_4^*\) \(=\) \(\ds 1385\) \(\ds = E_8\)
\(\ds E_5^*\) \(=\) \(\ds 50 \, 521\) \(\ds = -E_{10}\)
\(\ds E_6^*\) \(=\) \(\ds 2 \, 702 \, 765\) \(\ds = E_{12}\)
\(\ds E_7^*\) \(=\) \(\ds 199 \, 360 \, 981\) \(\ds = -E_{14}\)
\(\ds E_8^*\) \(=\) \(\ds 19 \, 391 \, 512 \, 145\) \(\ds = E_{16}\)
\(\ds E_9^*\) \(=\) \(\ds 2 \, 404 \, 879 \, 675 \, 441\) \(\ds = -E_{18}\)
\(\ds E_{10}^*\) \(=\) \(\ds 370 \, 371 \, 188 \, 237 \, 525\) \(\ds = E_{20}\)
\(\ds E_{11}^*\) \(=\) \(\ds 69 \, 348 \, 874 \, 393 \, 137 \, 901\) \(\ds = -E_{22}\)
\(\ds E_{12}^*\) \(=\) \(\ds 15 \, 514 \, 534 \, 163 \, 557 \, 086 \, 905\) \(\ds = E_{24}\)

where $E_2, E_4, \ldots$ are the standard form Euler numbers.

This sequence is A000364 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Sources