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Function Repository Resource:

MultiFactorial

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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<nk, 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]
Out[2]=

Evaluate numerically:

In[3]:=
ResourceFunction["MultiFactorial"][5.5, 4]
Out[3]=

Scope (4) 

Evaluate for large arguments:

In[4]:=
ResourceFunction["MultiFactorial"][10000, 3] // Short
Out[4]=
In[5]:=
N[%]
Out[5]=

Evaluate at negative integers:

In[6]:=
Table[ResourceFunction["MultiFactorial"][-n, 5], {n, 7}]
Out[6]=

Evaluate for complex arguments:

In[7]:=
ResourceFunction["MultiFactorial"][-0.1 + 0.2 I, 3]
Out[7]=

MultiFactorial threads elementwise over lists:

In[8]:=
ResourceFunction["MultiFactorial"][{2, 3, 5, 7}, 4]
Out[8]=

Properties and Relations (2) 

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

In[9]:=
Table[ResourceFunction["MultiFactorial"][n, 1] == n!, {n, 7}]
Out[9]=

MultiFactorial[n,2] is equivalent to Factorial2[n] (n!!):

In[10]:=
Table[ResourceFunction["MultiFactorial"][n, 2] == n!!, {n, 7}]
Out[10]=

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]:=
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Out[14]=

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