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a(n) = n! [x^n] LW(T(x)), where T(x) = -W(-x) Euler's tree function, W(x) is the Lambert W function, and LW(x) = W(-W(x))/(-W(x)) (A340473).
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%I #3 Jan 09 2021 07:51:59

%S 1,1,3,22,209,2756,43717,839686,18581425,470707192,13352676101,

%T 420875581754,14566375690297,549877190829604,22472783629465093,

%U 989043215802778966,46631075599107558113,2345376059569552767344,125350843842721213505029,7095169059445749303612946

%N a(n) = n! [x^n] LW(T(x)), where T(x) = -W(-x) Euler's tree function, W(x) is the Lambert W function, and LW(x) = W(-W(x))/(-W(x)) (A340473).

%p W := x -> LambertW(x): T := x -> -W(-x): LW := x -> W(-W(x))/(-W(x)):

%p ser := series(LW(T(x)), x, 24): seq(n!*coeff(ser, x, n), n=0..19);

%Y Cf. A340473, A097174, A177885, A207833, A227176.

%Y Cf. A000169, A000272.

%K nonn

%O 0,3

%A _Peter Luschny_, Jan 09 2021