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EulerE[n]

gives the Euler number TemplateBox[{n}, EulerE].

EulerE[n,x]

gives the Euler polynomial TemplateBox[{n, x}, EulerE2].

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Applications  
Properties & Relations  
Possible Issues  
Neat Examples  
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EulerE

EulerE[n]

gives the Euler number TemplateBox[{n}, EulerE].

EulerE[n,x]

gives the Euler polynomial TemplateBox[{n, x}, EulerE2].

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • The Euler polynomials satisfy the generating function relation 2e^(xt)/(e^t+1)=sum_(n=0)^(infty)TemplateBox[{n, x}, EulerE2](t^n/n!).
  • The Euler numbers are given by TemplateBox[{n}, EulerE]=2^nTemplateBox[{n, {1, /, 2}}, EulerE2].
  • EulerE automatically threads over lists.

Examples

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Basic Examples  (2)

First 10 EulerE numbers:

Wolfram Language code: Table[EulerE[k], {k, 0, 10}]

Euler polynomials:

Wolfram Language code: Table[EulerE[n, z], {n, 0, 5}]

Scope  (4)

EulerE threads elementwise over lists:

Wolfram Language code: EulerE[{2, 4, 6}]

Plot Euler polynomials:

Wolfram Language code: Plot[Evaluate[Table[EulerE[k, z], {k, 0, 5}]], {z, 0, 1}]

Simple exact values are generated automatically:

Wolfram Language code: EulerE[n, (1/2)]

TraditionalForm formatting:

Wolfram Language code: EulerE[n]//TraditionalForm
Wolfram Language code: EulerE[n, x]//TraditionalForm

Applications  (2)

Implement the Boole summation formula:

Wolfram Language code: BooleSummation[f_, {k_, 0, n_}, m_] := (1/2) Underoverscript[∑, j = 0, m - 1](EulerE[j, 0]/j!) ((-1)^n (Subscript[∂, {k, j}]f /. k -> n + 1) + (Subscript[∂, {k, j}]f /. k -> 0))

First a sequence of approximations to sum_(k=0)^n(-1)^kk^3:

Wolfram Language code: Table[BooleSummation[k ^ 3, {k, 0, n}, m], {m, 5}]//FullSimplify

The sequence converges to the exact answer:

Wolfram Language code: % /. n -> 10

Plot roots of Euler polynomials in the complex plane:

Wolfram Language code: Graphics[Table[{Hue[n / 30], Point[{Re[#], Im[#]}]& /@ (z /. NSolve[EulerE[n, z] == 0, z])}, {n, 30}]]

Properties & Relations  (5)

Find Euler numbers from their generating function:

Wolfram Language code: Table[Limit[D[Sech[z], {z, n}], z -> 0], {n, 10}]
Wolfram Language code: Table[EulerE[n], {n, 0, 10}]

Find Euler polynomials from their generating function:

Wolfram Language code: CoefficientList[Series[Exp[x t] / (Exp[t] + 1), {t, 0, 5}], t]Table[2n!, {n, 0, 5}]//Expand
Wolfram Language code: Table[EulerE[n, x], {n, 0, 5}]

EulerE can be represented as a DifferenceRoot:

Wolfram Language code: DifferenceRootReduce[EulerE[1, k], k]

FindSequenceFunction can recognize the EulerE sequence:

Wolfram Language code: Table[EulerE[n], {n, 10}]
Wolfram Language code: FindSequenceFunction[%, n]

The exponential generating function for EulerE:

Wolfram Language code: ExponentialGeneratingFunction[EulerE[n], n, x]

Possible Issues  (1)

Algorithmically produced results are often expressed using Zeta instead of EulerE:

Wolfram Language code: (2^n + 2 n! /π^n + 1)Underoverscript[∑, k = 0, ∞](1/(2 k + 1)^n + 1)Cos[k π - (n π/2)]
Wolfram Language code: Table[%, {n, 10}]//FullSimplify
Wolfram Language code: Table[EulerE[n], {n, 10}]

Neat Examples  (4)

Umbral calculus with Euler numbers:

Wolfram Language code: Expand[(e - I) ^ 10]
Wolfram Language code: % /. e ^ k_. :> Abs[EulerE[k]]

Histogram of digits of 10000^(th) Euler number:

Wolfram Language code: Histogram[IntegerDigits[EulerE[10000]]]

The sequence of Euler numbers modulo a fixed number is periodic:

Wolfram Language code: DiscretePlot[Mod[EulerE[2n], 17], {n, 50}]

Define a Hankel matrix whose entries are the Euler numbers:

Wolfram Language code: eulerHankel[n_] := HankelMatrix[EulerE[Range[0, n - 1]], EulerE[Range[n - 1, 2n - 2]]]
Wolfram Language code: eulerHankel[5]//MatrixForm

Its determinant can be expressed in terms of the Barnes G-function:

Wolfram Language code: Table[Det[eulerHankel[n]] == (-1)^Binomial[n, 2]BarnesG[n + 1]^2, {n, 10}]
Wolfram Research (1988), EulerE, Wolfram Language function, https://reference.wolfram.com/language/ref/EulerE.html.

Text

Wolfram Research (1988), EulerE, Wolfram Language function, https://reference.wolfram.com/language/ref/EulerE.html.

CMS

Wolfram Language. 1988. "EulerE." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/EulerE.html.

APA

Wolfram Language. (1988). EulerE. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/EulerE.html

BibTeX

@misc{reference.wolfram_2026_eulere, author="Wolfram Research", title="{EulerE}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/EulerE.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_eulere, organization={Wolfram Research}, title={EulerE}, year={1988}, url={https://reference.wolfram.com/language/ref/EulerE.html}, note=[Accessed: 13-June-2026]}