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AlternatingFactorial
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AlternatingFactorial

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

Details

Examples

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Basic Examples  (6)Summary of the most common use cases

Compute the first few alternating factorials:

In[1]:=1
✖
https://wolfram.com/xid/0mlcfc7ep4si-cfi83b
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0mlcfc7ep4si-4uocp
Out[1]=1

Plot over a subset of the complexes:

In[1]:=1
✖
https://wolfram.com/xid/0mlcfc7ep4si-b7f34n
Out[1]=1

Expand the alternating factorial in terms of other functions:

In[1]:=1
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https://wolfram.com/xid/0mlcfc7ep4si-v46voh
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0mlcfc7ep4si-jqvg93
Out[2]=2

Give the closed form of the following alternating sum:

In[1]:=1
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https://wolfram.com/xid/0mlcfc7ep4si-hmlp64
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0mlcfc7ep4si-p7ljho
In[2]:=2
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https://wolfram.com/xid/0mlcfc7ep4si-384vzm
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0mlcfc7ep4si-jlrdhp
Out[3]=3

Scope  (18)Survey of the scope of standard use cases

Numerical Evaluation  (6)

Evaluate numerically:

In[1]:=1
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https://wolfram.com/xid/0mlcfc7ep4si-itrtf
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0mlcfc7ep4si-cksbl4
Out[2]=2

Evaluate to high precision:

In[1]:=1
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https://wolfram.com/xid/0mlcfc7ep4si-b0wt9
Out[1]=1

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

In[2]:=2
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https://wolfram.com/xid/0mlcfc7ep4si-xth5g
Out[2]=2

AlternatingFactorial can take complex number inputs:

In[1]:=1
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https://wolfram.com/xid/0mlcfc7ep4si-hfml09
Out[1]=1

Evaluate efficiently at high precision:

In[1]:=1
✖
https://wolfram.com/xid/0mlcfc7ep4si-di5gcr
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0mlcfc7ep4si-bq2c6r
Out[2]=2

Compute the elementwise values of an array using automatic threading:

In[1]:=1
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https://wolfram.com/xid/0mlcfc7ep4si-thgd2
Out[1]=1

Or compute the matrix AlternatingFactorial function using MatrixFunction:

In[2]:=2
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https://wolfram.com/xid/0mlcfc7ep4si-o5jpo
Out[2]=2

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

In[1]:=1
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https://wolfram.com/xid/0mlcfc7ep4si-dj6d9x
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0mlcfc7ep4si-f892wm
Out[2]=2

Or compute average-case statistical intervals using Around:

In[3]:=3
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https://wolfram.com/xid/0mlcfc7ep4si-cw18bq
Out[3]=3

Specific Values  (3)

Values of AlternatingFactorial at fixed points:

In[1]:=1
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https://wolfram.com/xid/0mlcfc7ep4si-nww7l
Out[1]=1

Value at zero:

In[1]:=1
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https://wolfram.com/xid/0mlcfc7ep4si-e41pf2
Out[1]=1

Evaluate symbolically:

In[1]:=1
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https://wolfram.com/xid/0mlcfc7ep4si-eswwm
Out[1]=1

Visualization  (2)

Plot the absolute value of AlternatingFactorial:

In[1]:=1
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https://wolfram.com/xid/0mlcfc7ep4si-ecj8m7
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0mlcfc7ep4si-kgd8nu
Out[1]=1

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

In[2]:=2
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https://wolfram.com/xid/0mlcfc7ep4si-f3fwli
Out[2]=2

Function Properties  (7)

Real domain of AlternatingFactorial:

In[1]:=1
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https://wolfram.com/xid/0mlcfc7ep4si-cl7ele
Out[1]=1

Complex domain:

In[2]:=2
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https://wolfram.com/xid/0mlcfc7ep4si-de3irc
Out[2]=2

TraditionalForm formatting:

In[1]:=1
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https://wolfram.com/xid/0mlcfc7ep4si-efpvd0

AlternatingFactorial is not an analytic function:

In[1]:=1
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https://wolfram.com/xid/0mlcfc7ep4si-gva6yl
Out[1]=1

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

In[2]:=2
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https://wolfram.com/xid/0mlcfc7ep4si-fyfbxx
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0mlcfc7ep4si-5vh4e
Out[3]=3

AlternatingFactorial is neither nondecreasing nor nonincreasing:

In[1]:=1
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https://wolfram.com/xid/0mlcfc7ep4si-2ra8g
Out[1]=1

AlternatingFactorial is not injective:

In[1]:=1
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https://wolfram.com/xid/0mlcfc7ep4si-c9npzh
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0mlcfc7ep4si-b5buvp
Out[2]=2

AlternatingFactorial is neither non-negative nor non-positive:

In[1]:=1
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https://wolfram.com/xid/0mlcfc7ep4si-dvzykj
Out[1]=1

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

In[2]:=2
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https://wolfram.com/xid/0mlcfc7ep4si-estge4
Out[2]=2

AlternatingFactorial is neither convex nor concave:

In[1]:=1
✖
https://wolfram.com/xid/0mlcfc7ep4si-l0srvu
Out[1]=1

Applications  (1)Sample problems that can be solved with this function

AlternatingFactorial can be defined on the positive integers as follows:

In[1]:=1
✖
https://wolfram.com/xid/0mlcfc7ep4si-i19x6t
Out[1]=1

Verify the formula for a specific number:

In[2]:=2
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https://wolfram.com/xid/0mlcfc7ep4si-d1q8s3
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0mlcfc7ep4si-ot04rg
Out[3]=3
Wolfram Research (2014), AlternatingFactorial, Wolfram Language function, https://reference.wolfram.com/language/ref/AlternatingFactorial.html.
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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.

✖
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.

✖
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

✖
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: 29-March-2025 ]}

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@misc{reference.wolfram_2025_alternatingfactorial, author="Wolfram Research", title="{AlternatingFactorial}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/AlternatingFactorial.html}", note=[Accessed: 29-March-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: 29-March-2025 ]}

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@online{reference.wolfram_2025_alternatingfactorial, organization={Wolfram Research}, title={AlternatingFactorial}, year={2014}, url={https://reference.wolfram.com/language/ref/AlternatingFactorial.html}, note=[Accessed: 29-March-2025 ]}