Euler Number Calculator
Enter an index n to compute the Euler number Eₙ exactly via a recurrence, with a table of E0 through Eₙ.
Input
Compute the Euler number Eₙ exactly via a recurrence. Enter a non-negative integer n.
Non-negative integer (up to 200)
Result
Euler number E(4)
5
Index
4
Parity
Even
Table of E0 to E({n})
| Index | Euler number |
|---|---|
| E(0) | 1 |
| E(1) | 0 |
| E(2) | -1 |
| E(3) | 0 |
| E(4) | 5 |
How it works
- The Euler numbers Eₙ are the coefficients in the power-series expansion of the hyperbolic secant: 1/cosh(t) = Σ Eₙ tⁿ / n!.
- Odd-index Euler numbers are all 0, while even-index ones alternate in sign (E0=1, E2=−1, E4=5, E6=−61, E8=1385, E10=−50521 and so on).
- This tool uses the recurrence Σ C(2m,2k)·E₂ₖ = 0 (m≥1, E0=1), solving E₂ₘ = −Σ C(2m,2k)·E₂ₖ to obtain each even-index value exactly.
- Because the values grow quickly, the computation uses arbitrary-precision integers (BigInt) so the exact integer is shown without overflow.
- The input n must be a non-negative integer, and this tool supports values up to 200.
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Euler Number Calculator