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A360548 E.g.f. satisfies A(x) = x * exp( 2*(x + A(x)) ).

%I A360548 #18 Feb 16 2025 08:34:04
%S A360548 0,1,8,96,1792,46080,1511424,60325888,2837970944,153778913280,
%T A360548 9432255692800,646039266656256,48874810528235520,4047655951598092288,
%U A360548 364221261622538141696,35384754572803304325120,3691411033400626898796544,411569264258973944034361344
%N A360548 E.g.f. satisfies A(x) = x * exp( 2*(x + A(x)) ).
%H A360548 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A360548 E.g.f.: A(x) = (-1/2) * LambertW(-2*x * exp(2*x)).
%F A360548 a(n) = Sum_{k=1..n} (2*k)^(n-1) * binomial(n,k) = 4^(n-1) * A100526(n).
%F A360548 a(n) ~ sqrt(1 + LambertW(exp(-1))) * 2^(n-1) * n^(n-1) / (LambertW(exp(-1))^n * exp(n)). - _Vaclav Kotesovec_, Feb 17 2023
%p A360548 A360548 := proc(n)
%p A360548     add((2*k)^(n-1)*binomial(n,k),k=1..n) ;
%p A360548 end proc:
%p A360548 seq(A360548(n),n=0..60) ; # _R. J. Mathar_, Mar 12 2023
%o A360548 (PARI) my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(-2*x*exp(2*x))/2)))
%o A360548 (PARI) a(n) = sum(k=1, n, (2*k)^(n-1)*binomial(n, k));
%Y A360548 Cf. A100526, A216857, A360545.
%Y A360548 Cf. A038049, A201595, A214225, A360547.
%K A360548 nonn
%O A360548 0,3
%A A360548 _Seiichi Manyama_, Feb 11 2023