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XLeratorDB/statistics Documentation

SQL Server Stirling Number of the First Kind function


STIRLING1
Updated: 21 July 2017

Use the SQL Server scalar function STIRLING1 to calculate the Stirling numbers of the first kind. The Stirling numbers of the first kind are the coefficients in the expansion:

Where (x)n is the falling factorial (see FACTORIALPOWER).

Stirling numbers of the first kind exhibit the following recursive relationship.

Syntax
SELECT [wct].[STIRLING1] (
   <@n, int,>
  ,<@k, int,>)
Arguments
Input NameDefinition
@nNumber of items
@kNumber chosen
Return Type
float
Remarks
  • If @n <= 0 then NULL is returned.
  • If @k < 0 then NULL is returned.
  • If @k > n then NULL is returned.
  • s(0,0) = 1
  • s(n,0) = 0
  • Available in XLeratorDB / statistics 2008 only
Examples
Example #1

Calculate the coefficients for the expansion of: x(x-1)(x-2)(x-3)(x-4)

SELECT
     wct.STIRLING1(4,1) as [x]
    ,wct.STIRLING1(4,2) as [x^2]
    ,wct.STIRLING1(4,3) as [x^3]
    ,wct.STIRLING1(4,4) as [x^4]

This produces the following result.

The expanded equation would thus be: x4-6x3+11x2-6x

Example #2

A lower triangular representation of the first few Stirling numbers of the first kind.

SELECT [1],[2],[3],[4],[5],[6],[7],[8],[9],[10]
FROM (
    SELECT
         n.SeriesValue as n
        ,k.seriesValue as k
        ,cast(wct.STIRLING1(n.SeriesValue,k.SeriesValue) as int) as S1
    FROM
        wct.SeriesInt(1,10,NULL,NULL,NULL)n
    CROSS APPLY
        wct.SeriesInt(1,n.seriesvalue,NULL,NULL,NULL)k
    )d   
PIVOT (MAX(S1) FOR k IN ([1],[2],[3],[4],[5],[6],[7],[8],[9],[10]))pvt
ORDER BY
    n

This produces the following result.

See Also


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