cp's OEIS Frontend

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

%I A383130 #25 Jun 25 2025 00:36:58
%S A383130 1,1,1,5,17,133,1927,13582711,92612482895,10402118970990527,
%T A383130 59203666396198716260449,83631044830029201279016528831,
%U A383130 1149522186344339904123210420373026673,458029700061597358458976211208014885543904637441,203695852839150317577316770934832249000714992664672874100151
%N A383130 Coefficients of the linear terms in the continued fraction representation of the product logarithm.
%C A383130 The continued fraction only produces values for the principal branch of the product logarithm.
%H A383130 Cristina B. Corcino, Roberto B. Corcino, and István Mező, <a href="https://doi.org/10.1007/s00010-018-0559-2">Continued fraction expansions for the Lambert W function</a>, Aequat. Math. 93, 485-498 (2019)
%e A383130 LambertW(x) = x/(1 + x/(1 + x/(2 + 5*x/(3 + 17*x/(10 + 133*x/(17 + 1927*x/(190 + ... ))))))).
%t A383130 ClearAll[cf, x];
%t A383130 cf[ O[x]] = {};
%t A383130 cf[ a0_ + O[x]] := {a0};
%t A383130 cf[ ps_] := Module[ {a0, a1, u, v},
%t A383130   a0 = SeriesCoefficient[ ps, {x, 0, 0}];
%t A383130   a1 = SeriesCoefficient[ ps, {x, 0, 1}];
%t A383130   u = Numerator[a1]; v = Denominator[a1];
%t A383130   Join[ If[ a0==0, {}, {a0}],
%t A383130      Prepend[cf[ u*x/(ps-a0) - v], {u,v}]]];
%t A383130 (* Lambert W function W_0(x) up to O(x)^(M+1) *)
%t A383130 M = 10; W0 = Sum[ x^n*(-n)^(n-1)/n!, {n, 1, M}] + x*O[x]^M;
%t A383130 cf[W0] //InputForm
%t A383130 (* {{1, 1}, {1, 1}, {1, 2}, {5, 3}, {17, 10}, {133, 17},
%t A383130  {1927, 190}, {13582711, 94423}, {92612482895, 1597966},
%t A383130  {10402118970990527, 8773814169}} *)
%t A383130 (* Note: Change M to the number of terms to be generated *)
%Y A383130 Cf. A213236 (e.g.f. of LambertW).
%K A383130 nonn
%O A383130 1,4
%A A383130 _Jacob DeMoss_, Jun 17 2025
%E A383130 More terms from _Alois P. Heinz_, Jun 17 2025