Wikipedia's image for the representation
I was just wondering if anyone has an extended version of it. (Like a longer fraction so it is more accurate)
I need to get further than the 94423 term in
x/(1+x/(1+x/(2+3x/(3+17x/(10+1927x/(190+13582711x/(94423 ...
On
The Mathematica code here does what you want.
ClearAll[cf, x];
cf[ O[x]] = {};
cf[ a0_ + O[x]] := {a0};
cf[ ps_] := Module[ {a0, a1, u, v},
a0 = SeriesCoefficient[ ps, {x, 0, 0}];
a1 = SeriesCoefficient[ ps, {x, 0, 1}];
u = Numerator[a1]; v = Denominator[a1];
Join[ If[ a0==0, {}, {a0}],
Prepend[cf[ u*x/(ps-a0) - v], {u,v}]]];
(* Lambert W function W_0(x) up to O(x)^(M+1) *)
M = 10; W0 = Sum[ x^n*(-n)^(n-1)/n!, {n, 1, M}] + x*O[x]^M;
cf[W0] //InputForm
(* {{1, 1}, {1, 1}, {1, 2}, {5, 3}, {17, 10}, {133, 17},
{1927, 190}, {13582711, 94423}, {92612482895, 1597966},
{10402118970990527, 8773814169}} *)
Where the M = 10 can be set to how many terms you want.
According to Maple, it's $$\frac{a_1 x}{b_1 + \frac{a_2 x}{b_2 + \ldots}}$$ where $[a_i, b_i]$ are $$ [1, 1], [1, 1], [1, 2], [5, 3], [17, 10], [133, 17], [1927, 190], [13582711, 94423], [92612482895 , 1597966], [10402118970990527 , 8773814169], [59203666396198716260449 ,10796179523602],\ldots$$
The $a_i$ and $b_i$ seem not to be in the OEIS.