A383130 Coefficients of the linear terms in the continued fraction representation of the product logarithm.

1, 1, 1, 5, 17, 133, 1927, 13582711, 92612482895, 10402118970990527, 59203666396198716260449, 83631044830029201279016528831, 1149522186344339904123210420373026673, 458029700061597358458976211208014885543904637441, 203695852839150317577316770934832249000714992664672874100151

Offset: 1

Author

Jacob DeMoss, Jun 17 2025

Keywords

• nonn

Comments

The continued fraction only produces values for the principal branch of the product logarithm.

Examples

			LambertW(x) = x/(1 + x/(1 + x/(2 + 5*x/(3 + 17*x/(10 + 133*x/(17 + 1927*x/(190 + ... ))))))).
		

Links

• Cristina B. Corcino, Roberto B. Corcino, and István Mező, Continued fraction expansions for the Lambert W function, Aequat. Math. 93, 485-498 (2019)

Crossrefs

Cf. A213236 (e.g.f. of LambertW).

Programs

Mathematica

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}} *)
(* Note: Change M to the number of terms to be generated *)

Extensions

More terms from Alois P. Heinz, Jun 17 2025