STIRLING1

Updated 2024-03-14 20:23:52.863000

Syntax

SELECT [westclintech].[wct].[STIRLING1] (
   <@n, int,>
  ,<@k, int,>)

Description

Use the 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 of the falling factorial:

(x)_n = x(x-1)(x-2)\cdots(x-n+1)

Into the powers of the variable x;

(x)_n = \sum_{k=0}^n s(n,k) x^k

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

s(n,k)=s(n-1,k-1)-(n-1)s(n-1,k)

Arguments

@n

Number of items.

@k

Number 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.

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.

{"columns":[{"field":"x","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"},{"field":"x^2","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"},{"field":"x^3","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"},{"field":"x^4","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"}],"rows":[{"x":"-6","x^2":"11","x^3":"-6","x^4":"1"}]}

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;