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E.g.f.: exp(2*x)/(1+LambertW(-x)).
3

%I #13 Nov 08 2017 02:30:58

%S 1,3,12,71,616,7197,105052,1829291,36922928,846851993,21744781684,

%T 617832652527,19242299657896,651815827343189,23857403245171724,

%U 938247816632341043,39455261828928309088,1766645684585351990961,83913998998426051745764,4214295288128637488870327,223120214856875472660345176

%N E.g.f.: exp(2*x)/(1+LambertW(-x)).

%H G. C. Greubel, <a href="/A277457/b277457.txt">Table of n, a(n) for n = 0..385</a>

%F a(n) ~ exp(2*exp(-1)) * n^n.

%t CoefficientList[Series[Exp[2*x]/(1+LambertW[-x]), {x, 0, 20}], x]*Range[0, 20]!

%t Table[1 + Sum[Binomial[n, m]*(1 + Sum[Binomial[m, k]*k^k, {k, 1, m}]), {m, 1, n}], {n, 0, 20}]

%t Table[2^n + Sum[Binomial[n, k]*2^(n-k)*k^k, {k, 1, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Oct 28 2016 *)

%o (PARI) x='x+O('x^50); Vec(serlaplace(exp(2*x)/(1 + lambertw(-x)))) \\ _G. C. Greubel_, Nov 07 2017

%Y Cf. A086331, A277454, A277456, A277485.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Oct 16 2016