While toying with the factorial function, I stumbled with the following definition:(adsbygoogle = window.adsbygoogle || []).push({});

Let [tex]P_k^{\,n}[/tex] be the sum of all possible products of k different factors, taken from {1,2,3,...n}.

For example, [tex]P_3^{\,4}[/tex] = 1.2.3 + 1.2.4 + 1.3.4 + 2.3.4 = 50

Now, according to the Sloane site, this guys turn to be the (unsigned) Stirling numbers of the first kind (with the indexes shifted; actually, [tex]P_k^{\,n}[/tex] = |S1(n+1,n+1-k)|), but I fail to see an immediate relation between the above statement and the usual definition.

A possible route might involve proving that "my" numbers follow the recurrence relation |S1(n,k)| = |S1(n-1,k-1)| + n . |S1(n-1,k)|.

But, before I start scribbling (I'm lazy), can somebody provide a hint, or a link? Thanks! Contribute to avoid the reinvention of the wheel!

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# Unsigned Stirling numbers of the first kind

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