Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 MultiFactorial

Source Notebook

Evaluate the multifactorial function

Contributed by: Jan Mangaldan

ResourceFunction["MultiFactorial"][n,k]

gives the kth multifactorial of n.

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
For integer n, the kth multifactorial of nn!(k) is equal to n if 0<n≤k, and n ((n-k)!(k)) if k<n.
For certain special arguments, ResourceFunction["MultiFactorial"] automatically evaluates to exact values.
ResourceFunction["MultiFactorial"] can be evaluated to arbitrary numerical precision.
ResourceFunction["MultiFactorial"] automatically threads over lists.

Examples

Basic Examples (2) 

Integer values:

In[1]:=
Table[ResourceFunction["MultiFactorial"][n, 3], {n, 10}]
Out[1]=
In[2]:=
ResourceFunction["MultiFactorial"][50, 4]
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Evaluate numerically:

In[3]:=
ResourceFunction["MultiFactorial"][5.5, 4]
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Scope (4) 

Evaluate for large arguments:

In[4]:=
ResourceFunction["MultiFactorial"][10000, 3] // Short
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In[5]:=
N[%]
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Evaluate at negative integers:

In[6]:=
Table[ResourceFunction["MultiFactorial"][-n, 5], {n, 7}]
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Evaluate for complex arguments:

In[7]:=
ResourceFunction["MultiFactorial"][-0.1 + 0.2 I, 3]
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MultiFactorial threads elementwise over lists:

In[8]:=
ResourceFunction["MultiFactorial"][{2, 3, 5, 7}, 4]
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Properties and Relations (2) 

MultiFactorial[n,1] is equivalent to Factorial[n] (n!):

In[9]:=
Table[ResourceFunction["MultiFactorial"][n, 1] == n!, {n, 7}]
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MultiFactorial[n,2] is equivalent to Factorial2[n] (n!!):

In[10]:=
Table[ResourceFunction["MultiFactorial"][n, 2] == n!!, {n, 7}]
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Possible Issues (2) 

Large arguments can give results too large to be computed explicitly, even approximately:

In[11]:=
ResourceFunction["MultiFactorial"][10.^13, 4]
Out[11]=

Machine-number inputs can give arbitrary‐precision results:

In[12]:=
ResourceFunction["MultiFactorial"][10.^6, 5]
Out[12]=
In[13]:=
MachineNumberQ[%]
Out[13]=

Neat Examples (1) 

Evaluate various reciprocal multifactorial constants:

In[14]:=
Table[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(n = 0\), \(\[Infinity]\)]\* FractionBox["1", RowBox[{ InterpretationBox[ TagBox[ DynamicModuleBox[{Typeset`open = False}, FrameBox[ PaneSelectorBox[{False->GridBox[{ { PaneBox[GridBox[{ { StyleBox[ StyleBox[ AdjustmentBox["\<\"[\[FilledSmallSquare]]\"\>", BoxBaselineShift->-0.25, BoxMargins->{{0, 0}, {-1, -1}}], "ResourceFunctionIcon", FontColor->RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784]], ShowStringCharacters->False, FontFamily->"Source Sans Pro Black", FontSize->0.6538461538461539 Inherited, FontWeight->"Heavy", PrivateFontOptions->{"OperatorSubstitution"->False}], StyleBox[ RowBox[{ StyleBox["MultiFactorial", "ResourceFunctionLabel", FontFamily->"Source Sans Pro"], " "}], ShowAutoStyles->False, ShowStringCharacters->False, FontSize->Rational[12, 13] Inherited, FontColor->GrayLevel[0.1]]} }, GridBoxSpacings->{"Columns" -> {{0.25}}}], Alignment->Left, BaseStyle->{LineSpacing -> {0, 0}, LineBreakWithin -> False}, BaselinePosition->Baseline, FrameMargins->{{3, 0}, {0, 0}}], ItemBox[ PaneBox[ TogglerBox[Dynamic[Typeset`open], {True-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeCloser"], ImageSizeCache->{11., {1., 10.}}], False-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeOpener"], ImageSizeCache->{11., {1., 10.}}]}, Appearance->None, BaselinePosition->Baseline, ContentPadding->False, FrameMargins->0], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{1, 1}, {0, 0}}], Frame->{{ RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627, 0.5], False}, {False, False}}]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings->{"Columns" -> {{0}}, "Rows" -> {{0}}}], True->GridBox[{ {GridBox[{ { PaneBox[GridBox[{ { StyleBox[ StyleBox[ AdjustmentBox["\<\"[\[FilledSmallSquare]]\"\>", BoxBaselineShift->-0.25, BoxMargins->{{0, 0}, {-1, -1}}], "ResourceFunctionIcon", FontColor->RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784]], ShowStringCharacters->False, FontFamily->"Source Sans Pro Black", FontSize->0.6538461538461539 Inherited, FontWeight->"Heavy", PrivateFontOptions->{"OperatorSubstitution"->False}], StyleBox[ RowBox[{ StyleBox["MultiFactorial", "ResourceFunctionLabel", FontFamily->"Source Sans Pro"], " "}], ShowAutoStyles->False, ShowStringCharacters->False, FontSize->Rational[12, 13] Inherited, FontColor->GrayLevel[0.1]]} }, GridBoxSpacings->{"Columns" -> {{0.25}}}], Alignment->Left, BaseStyle->{LineSpacing -> {0, 0}, LineBreakWithin -> False}, BaselinePosition->Baseline, FrameMargins->{{3, 0}, {0, 0}}], ItemBox[ PaneBox[ TogglerBox[Dynamic[Typeset`open], {True-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeCloser"]], False-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeOpener"]]}, Appearance->None, BaselinePosition->Baseline, ContentPadding->False, FrameMargins->0], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{1, 1}, {0, 0}}], Frame->{{ RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627, 0.5], False}, {False, False}}]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings->{"Columns" -> {{0}}, "Rows" -> {{0}}}]}, { StyleBox[ PaneBox[GridBox[{ { RowBox[{ TagBox["\<\"Version (latest): \"\>", "IconizedLabel"], " ", TagBox["\<\"1.0.1\"\>", "IconizedItem"]}]}, { TagBox[ TemplateBox[{"\"Documentation »\"", "https://resources.wolframcloud.com/FunctionRepository/resources/MultiFactorial"}, "HyperlinkURL"], "IconizedItem"]} }, DefaultBaseStyle->"Column", GridBoxAlignment->{"Columns" -> {{Left}}}, GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{5, 4}, {0, 4}}], "DialogStyle", FontFamily->"Roboto", FontSize->11]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxDividers->{"Columns" -> {{None}}, "Rows" -> {False, { GrayLevel[0.8]}, False}}, GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}]}, Dynamic[Typeset`open], BaselinePosition->Baseline, ImageSize->Automatic], Background->RGBColor[ 0.9686274509803922, 0.9764705882352941, 0.984313725490196], BaselinePosition->Baseline, DefaultBaseStyle->{}, FrameMargins->{{0, 0}, {1, 0}}, FrameStyle->RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627], RoundingRadius->4]], {"FunctionResourceBox", RGBColor[0.8745098039215686, 0.2784313725490196, 0.03137254901960784], "MultiFactorial"}, TagBoxNote->"FunctionResourceBox"], ResourceFunction["MultiFactorial"], BoxID -> "MultiFactorial", Selectable->False], "[", RowBox[{"n", ",", "k"}], "]"}]]\), {k, {1, 2, 4}}]
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Version History

  • 1.0.1 – 01 March 2021
  • 1.0.0 – 08 February 2021

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