easy pieces : E^-((-x + 1)^y - 1)

[Wouter Meeussen]

the 

Series[ E^-((-x + 1)^y - 1) , {x,0, W}, {y,0, W } ]

is just  ('cause A000587 makes it easy)

1+Sum[  
        Sum[(-1)^(n + k)  StirlingS2[m, k], {k, m}] *
           StirlingS1[n,m] y^m x^n / n! 
  , {n, 0, W }, {m, 0, n}]


or, the 2-D coefficient list  :


{1}
{0, 1}
{0, 1}
{0, 2, 0, -1}
{0, 6, 0, -6, 1}
{0, 24, 0, -35, 10, 2}
{0, 120, 0, -225, 85, 30, -9}
{0, 720, 0, -1624, 735, 350, -189, 9}
{0, 5040, 0, -13132, 6769, 3920, -2898, 252, 50}
{0, 40320, 0, -118124, 67284, 44898, -40824, 4914, 1800, -267}
{0, 362880, 0, -1172700, 723680, 538650, -569457, 85050, 43500, -12015, 413}
{0, 3628800, 0, -12753576, 8409500, 6833860, -8118495, 1419957, 907500,
-352440, 22715, 2180}
{0, 39916800, 0, -150917976, 105258076, 91991460, -120055815, 23738022,
17871150, -8722890, 795025, 143880, -17731}



*************************************************************************
little correction to A000587 :

" Formula:   |a(n)| = |S(n,1)-S(n,2)+...+c*S(n,n)|,where c=(-1)^(n+1) "
should be summed from zero, with a minus up front, and c defined using the
running index k, not n.
" Formula:   |a(n)| = | Sum( (k=0 to n) (-1)^(k+1) S(n,k) ) | " ,

in Mma :
Table[-1 * Sum[(-1)^( k + 1)  StirlingS2[n, k], {k, 0, n}], {n, 0, 12}]

{1, -1, 0, 1, 1, -2, -9, -9, 50, 267, 413, -2180, -17731}
Dr. Wouter L. J. MEEUSSEN
w.meeussen.vdmcc at vandemoortele.be
eu000949 at pophost.eunet.be