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%I #22 Oct 13 2022 13:26:53
%S 1,8,648,82944,1296000000,69984000000,403443833184000000,
%T 26440095051546624000000,42154051662866968215552000000,
%U 263462822892918551347200000000000,826859199578154686310659783668531200000000000
%N Denominator values occurring in formulas for the n-th integration of the Lambert W function.
%e If we use J(n, f(x)) notation for the n-th integration, then we can find the denominator
%e J(1, W(x)) = x/(W(x) (W(x)^2 - W(x) + 1) + c
%e J(2, W(x)) = x^2/(8W(x)^2) (4W(x)^3 - 6W(x)^2 + 6W(x) + 1) + kx + c
%e J(3, W(x)) = x^3/(648W(x)^3) (108W(x)^4 - 198W(x)^3 + 198W(x)^2 + 57W(x) + 8) + x^2/2 h + kx + c
%e ...
%e where c, k, h are constants.
%t max = 10; Table[Denominator[Together[Rest[NestList[Integrate[#, x] &, LambertW[x], max]]]][[k]] / ProductLog[x]^k, {k, 1, max}] (* _Vaclav Kotesovec_, Apr 14 2022 *)
%Y Cf. A352592.
%K nonn
%O 1,2
%A _Lukáš Backa_, Mar 21 2022
%E More terms from _Vaclav Kotesovec_, Apr 14 2022
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