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Stirling1

computes the Stirling numbers of the first kind

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

Stirling1(n, m)

combinat[stirling1](n, m)

Parameters

n, m

-

integers

Description

• 

The Stirling1(n,m) command computes the Stirling numbers of the first kind using the (implicit) generating function

m=0nStirling1n,mxm=−1n+1ΓnxΓx=xx1...xn+1

  

Instead of Stirling1 you can also use the synonym combinat[stirling1].

• 

Regarding combinatorial functions, 1nmStirling1n,m is the number of permutations of n symbols that have exactly m cycles. The Stirling numbers also enter binomial series, Mathieu function formulas, and are relevant in physical applications.

• 

The Stirling numbers of the first kind can be expressed as an explicit Sum with the Stirling numbers of second kind in the coefficients:

Stirling1n,m=k=0nm−1kn1+knm+k2nmnmkStirling2nm+k,k

  

Since the Stirling numbers of the second kind also admit an explicit Sum representation,

Stirling2m,n=k=0nnkkmn!−1kn

  

then, an explicit double Sum representation for Stirling1 is possible by combining the two formulas above. (See the Examples section.)

Examples

Stirling1 only evaluates to a number when m and n are positive integers

Stirling1m,n

Stirling1m,n

(1)

=convert,Sum

Stirling1m,n=_k1=0mn_k2=0_k112_k1_k2binomialm1+_k1,mn+_k1binomial2mn,mn_k1binomial_k1,_k2_k2mn+_k1_k1!

(2)

eval,m=10,n=5

269325=_k1=05_k2=0_k112_k1_k2binomial9+_k1,5+_k1binomial15,5_k1binomial_k1,_k2_k25+_k1_k1!

(3)

value

269325=269325

(4)

See Also

combinat

Stirling2

 


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