[some loose ideas]

[Christian G.Bower]

Wouter Meeussen <eu000949 at pophost.eunet.be> wrote:
> hi sequers,
> 
> 
> Series of  E^(1/k^z - 1) for k around k=1 
> equals    Sum[ Poly[j,z] / j!  (k-1) ^j , {j, 0, \[Infinity]}]
> 
> Poly[j,z] stands for a polynome in z of degree j :
> {1,
>    -z,
>     z +   2 z^2,
>  -2 z -   6 z^2 -   5 z^3, 
>   6 z +  22 z^2 +  30 z^3 +  15 z^4,
> -24 z - 100 z^2 - 175 z^3 - 150 z^4 - 52 z^5}
> 
> The Coefficient List table, after transforming z->-z, is:
> 
> {1}
> {0, 1}
> {0, -1, 2}
> {0, 2, -6, 5}
> {0, -6, 22, -30, 15}
> {0, 24, -100, 175, -150, 52}
> {0, -120, 548, -1125, 1275, -780, 203}
> {0, 720, -3528, 8120, -11025, 9100, -4263, 877}
> {0, -5040, 26136, -65660, 101535, -101920, 65366, -24556, 4140}
> {0, 40320, -219168, 590620, -1009260, 1167348, -920808, 478842, -149040,
21147}
> {0, -362880, 2053152, -5863500, 10855200, -14004900, 12844419, -8287650,
> 3601800, -951615, 115975}
> {0, 3628800, -21257280, 63767880, -126142500, 177680360, -183117165,
> 138366921, -75141000, 27914040, -6378625, 678570}
> 

I get this triangle by taking e^(Stirling-1 triangle)

Stirling-1 has e.g.f. (1+x)^y
So this triangle has e.g.f. exp((1+x)^y - 1).

The absolute values represent the number of "sets of permutations" of n
objects so that the total (cumulative) number of cycles in the sets is k.

Christian


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