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AlternatingFactorial[n]

gives the alternating factorial TemplateBox[{n}, AlternatingFactorial].

Details
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Examples  
Basic Examples  
Scope  
Numerical Evaluation  
Specific Values  
Visualization  
Function Properties  
Applications  
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AlternatingFactorial

AlternatingFactorial[n]

gives the alternating factorial TemplateBox[{n}, AlternatingFactorial].

Details

Examples

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Basic Examples  (6)

Compute the first few alternating factorials:

Wolfram Language code: Table[AlternatingFactorial[k], {k, 12}]

Plot the values on a log scale over a subset of the reals:

Wolfram Language code: DiscretePlot[AlternatingFactorial[k], {k, 12}, ScalingFunctions -> "Log"]

Plot over a subset of the complexes:

Wolfram Language code: ComplexPlot3D[AlternatingFactorial[z], {z, -2 - 2I, 2 + 2I}, PlotLegends -> Automatic]

Expand the alternating factorial in terms of other functions:

Wolfram Language code: FunctionExpand[AlternatingFactorial[n]]
Wolfram Language code: FullSimplify[%, Element[n, Integers]]

Give the closed form of the following alternating sum:

Wolfram Language code: Sum[(-1) ^ (n - k)k!, {k, n}]

The alternating factorial numbers give the solution to the following recurrence:

Wolfram Language code: a[1] = 1; a[n_] := a[n] = n! - a[n - 1];
Wolfram Language code: Array[a, 12]
Wolfram Language code: AlternatingFactorial[Range[12]]

Scope  (18)

Numerical Evaluation  (6)

Evaluate numerically:

Wolfram Language code: AlternatingFactorial[1.2]
Wolfram Language code: AlternatingFactorial[35]

Evaluate to high precision:

Wolfram Language code: N[AlternatingFactorial[(3/2)], 10]

The precision of the output tracks the precision of the input:

Wolfram Language code: AlternatingFactorial[1.20000000000000000000000]

AlternatingFactorial can take complex number inputs:

Wolfram Language code: AlternatingFactorial[2.5 + I]

Evaluate efficiently at high precision:

Wolfram Language code: AlternatingFactorial[45.`100]//Timing
Wolfram Language code: AlternatingFactorial[12.`5000];// Timing

Compute the elementwise values of an array using automatic threading:

Wolfram Language code: AlternatingFactorial[{{0, -1}, {-1, 0}}]

Or compute the matrix AlternatingFactorial function using MatrixFunction:

Wolfram Language code: MatrixFunction[AlternatingFactorial, {{0, -1}, {-1, 0}}]

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Wolfram Language code: AlternatingFactorial[Interval[{1.45, 1.46}]]
Wolfram Language code: AlternatingFactorial[CenteredInterval[-1.2, 0.001]]

Or compute average-case statistical intervals using Around:

Wolfram Language code: AlternatingFactorial[ Around[2, 0.01]]

Specific Values  (3)

Values of AlternatingFactorial at fixed points:

Wolfram Language code: Table[AlternatingFactorial[x ], {x, 1, 4}]

Value at zero:

Wolfram Language code: AlternatingFactorial[0]

Evaluate symbolically:

Wolfram Language code: AlternatingFactorial[x]//FunctionExpand

Visualization  (2)

Plot the absolute value of AlternatingFactorial:

Wolfram Language code: Plot[Abs[AlternatingFactorial[x]], {x, -1, 5}]

Plot the real part of TemplateBox[{z}, AlternatingFactorial]:

Wolfram Language code: ComplexContourPlot[Re[AlternatingFactorial[z]], {z, -3 - 3 I, 3 + 3I}, Contours -> 20]

Plot the imaginary part of TemplateBox[{z}, AlternatingFactorial]:

Wolfram Language code: ComplexContourPlot[Im[AlternatingFactorial[z]], {z, -3 - 3 I, 3 + 3I}, Contours -> 20]

Function Properties  (7)

Real domain of AlternatingFactorial:

Wolfram Language code: FunctionDomain[AlternatingFactorial[x], x]

Complex domain:

Wolfram Language code: FunctionDomain[AlternatingFactorial[z], z, Complexes]

TraditionalForm formatting:

Wolfram Language code: AlternatingFactorial[n]//TraditionalForm

AlternatingFactorial is not an analytic function:

Wolfram Language code: FunctionAnalytic[AlternatingFactorial[z], z]

AlternatingFactorial has both singularity and discontinuity for z≤-2:

Wolfram Language code: FunctionSingularities[AlternatingFactorial[z], z]
Wolfram Language code: FunctionDiscontinuities[AlternatingFactorial[z], z]

AlternatingFactorial is neither nondecreasing nor nonincreasing:

Wolfram Language code: FunctionMonotonicity[AlternatingFactorial[z], z]

AlternatingFactorial is not injective:

Wolfram Language code: FunctionInjective[AlternatingFactorial[z], z]
Wolfram Language code: Plot[{AlternatingFactorial[z], .8}, {z, -3, 3}]

AlternatingFactorial is neither non-negative nor non-positive:

Wolfram Language code: FunctionSign[AlternatingFactorial[z], z]

It is non-negative on the non-negative reals:

Wolfram Language code: FunctionSign[{AlternatingFactorial[z], z > 0}, z]

AlternatingFactorial is neither convex nor concave:

Wolfram Language code: FunctionConvexity[AlternatingFactorial[z], z]

Applications  (1)

AlternatingFactorial can be defined on the positive integers as follows:

Wolfram Language code: Sum[(-1) ^ (n - i) * i!, {i, 1, n}]

Verify the formula for a specific number:

Wolfram Language code: Sum[(-1) ^ (5 - i) * i!, {i, 1, 5}]
Wolfram Language code: AlternatingFactorial[5]
Wolfram Research (2014), AlternatingFactorial, Wolfram Language function, https://reference.wolfram.com/language/ref/AlternatingFactorial.html.

Text

Wolfram Research (2014), AlternatingFactorial, Wolfram Language function, https://reference.wolfram.com/language/ref/AlternatingFactorial.html.

CMS

Wolfram Language. 2014. "AlternatingFactorial." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/AlternatingFactorial.html.

APA

Wolfram Language. (2014). AlternatingFactorial. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AlternatingFactorial.html

BibTeX

@misc{reference.wolfram_2026_alternatingfactorial, author="Wolfram Research", title="{AlternatingFactorial}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/AlternatingFactorial.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_alternatingfactorial, organization={Wolfram Research}, title={AlternatingFactorial}, year={2014}, url={https://reference.wolfram.com/language/ref/AlternatingFactorial.html}, note=[Accessed: 12-June-2026]}