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



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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:

AlternatingFactorial threads elementwise over lists and matrices:

AlternatingFactorial can be used with Interval and CenteredInterval objects:

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,


Wolfram Research (2014), AlternatingFactorial, Wolfram Language function,


Wolfram Language. 2014. "AlternatingFactorial." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2014). AlternatingFactorial. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_alternatingfactorial, author="Wolfram Research", title="{AlternatingFactorial}", year="2014", howpublished="\url{}", note=[Accessed: 18-May-2024 ]}


@online{reference.wolfram_2024_alternatingfactorial, organization={Wolfram Research}, title={AlternatingFactorial}, year={2014}, url={}, note=[Accessed: 18-May-2024 ]}