Sum[(-n)^(n-1)/n! z^n,{n,\[Inf]}] == ProductLog[z]

[Wouter Meeussen]

this was a message in reply to
Bill Dubuque <wgd at martigny.ai.mit.edu>
with cc to [SeqFan], but I erroneously sent it to Sequences-Lookup instead.
So, here comes a copy to SeqFan:

Bill,

I take it that Lambert_W is the same as what Mathematica calls "ProductLog".
(quote:  ProductLog[z] gives the principal solution for w in  z = w  e^w  )

I most certainly find its Series development Weird :

z - z^2 + (3*z^3)/2 - (8*z^4)/3 + (125*z^5)/24 - (54*z^6)/5 + (16807*z^7)/720
 - (16384*z^8)/315 + (531441*z^9)/4480 - (156250*z^10)/567 + 
  (2357947691*z^11)/3628800 - (2985984*z^12)/1925

the Taylor Series coefficients are (the 'a_n '  in   a_n /n!  z^n    )


In[ ]:=Range[24]!  ( List@@Normal[Series[ProductLog[z],{z,0,24}]]/.z->1 )
Out[ ]=
{1,-2,9,-64,625,-7776,117649,-2097152,43046721,-1000000000,
  25937424601,-743008370688,23298085122481,-793714773254144,
  29192926025390625,-1152921504606846976,
  48661191875666868481,-2185911559738696531968,
  104127350297911241532841,-5242880000000000000000000,
  278218429446951548637196401,-15519448971100888972574851072,
  907846434775996175406740561329,-55572324035428505185378394701824}

This is Weird ! it's (-n)^(n-1)
*************************************************************************
this means that

Sum[(-n)^(n-1)/n! z^n,{n,\[Infinity]}] == ProductLog[z]

*************************************************************************

they are of course the limit of the n'th derivative of W at z->0.
This produces a "Lambert-triangle" after multiplying with (z/W)^n (1+W)^(2n+1):
(the first column are the Taylor Coefficients, the last are factorials)

  1,
 -2 - W,
 9 + 8 W + 2 W^2,
 -64 - 79 W - 36 W^2 - 6 W^3, 
 625 + 974 W + 622 W^2 + 192 W^3 + 24 W^4

etc. 

it= Table[Simplify[(Evaluate[D[ProductLog[z],{z,n}]]/.ProductLog[z]->W)*z^n/W^
n (1+W)^(2n-1)],{n,12}]   

Apply[List,it,1]/.W->1 


{1,
{-2,-1},
{9,8,2},
{-64,-79,-36,-6},
{625,974,622,192,24},
{-7776,-14543,-11758,-5126,-1200,-120},
{117649,255828,248250,137512,45756,8640,720},
{-2097152,-5187775,-5846760,-3892430,-1651480,-445572,-70560,-5040},
{43046721,119214746,152606870,118016760,60289700,20808776,4728816,645120,40320},
{-1000000000,-3062575399,-4381447314,-3846688510,-2284480360,-951218700,-277
339464,-54494064,-6531840,-362880},
{25937424601,86989216544,137384765938,134690452960,90822800600,44010015616,1
5509536224,3920009328,678843360,72576000,3628800},
{-743008370688,-2707207851599,-4674353001244,-5053964358758,-3804583321280,-
2094264010888,-859918048304,-263174384784,-58759159248,-9100236960,-87816960
0,-39916800}}



%I A000000 
%S A000000 {1},{2, 1},{9, 8, 2},{64, 79, 36, 6},{625, 974, 622, 192, 24},
%T A000000 {7776, 14543, 11758, 5126, 1200, 120},
%U A000000 {117649, 255828, 248250, 137512, 45756, 8640, 720}
%N A000000 polynomial in W generated by the nth derivative of W[z]=LambertW[
z]=ProductLog[z]
%R A000000 
%A A000000 w.meeussen.vdmcc at vandemoortele.be
%O A000000 0,1
%t A000000 Table[Simplify[(Evaluate[D[ProductLog[z],{z,n}]]/.ProductLog[z]->W)
 z^n/W^n (1+W)^(2n-1)],{n,12}]
%K A000000 nonn
Dr. Wouter L. J. MEEUSSEN
w.meeussen.vdmcc at vandemoortele.be
eu000949 at pophost.eunet.be