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:

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

Plot over a subset of the complexes:

Expand the alternating factorial in terms of other functions:

Give the closed form of the following alternating sum:

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

Scope  (18)

Numerical Evaluation  (6)

Evaluate numerically:

Evaluate to high precision:

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

AlternatingFactorial can take complex number inputs:

Evaluate efficiently at high precision:

Compute the elementwise values of an array using automatic threading:

Or compute the matrix AlternatingFactorial function using MatrixFunction:

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

Or compute average-case statistical intervals using Around:

Specific Values  (3)

Values of AlternatingFactorial at fixed points:

Value at zero:

Evaluate symbolically:

Visualization  (2)

Plot the absolute value of AlternatingFactorial:

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

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

Function Properties  (7)

Real domain of AlternatingFactorial:

Complex domain:

TraditionalForm formatting:

AlternatingFactorial is not an analytic function:

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

AlternatingFactorial is neither nondecreasing nor nonincreasing:

AlternatingFactorial is not injective:

AlternatingFactorial is neither non-negative nor non-positive:

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

AlternatingFactorial is neither convex nor concave:

Applications  (1)

AlternatingFactorial can be defined on the positive integers as follows:

Verify the formula for a specific number:

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_2025_alternatingfactorial, author="Wolfram Research", title="{AlternatingFactorial}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/AlternatingFactorial.html}", note=[Accessed: 20-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_alternatingfactorial, organization={Wolfram Research}, title={AlternatingFactorial}, year={2014}, url={https://reference.wolfram.com/language/ref/AlternatingFactorial.html}, note=[Accessed: 20-April-2025 ]}