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A360466 E.g.f. satisfies A(x) = exp(2 * x * exp(x) * A(x)).

%I A360466 #16 Feb 16 2025 08:34:04
%S A360466 1,2,16,206,3832,93962,2871820,105355406,4515648784,221598121490,
%T A360466 12257187851284,754703476252310,51204818674338328,3796079000648275226,
%U A360466 305328667748448560668,26483633169003911205278,2464307301750079915255840,244872778601760932275686434
%N A360466 E.g.f. satisfies A(x) = exp(2 * x * exp(x) * A(x)).
%H A360466 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A360466 E.g.f.: A(x) = exp( -LambertW(-2 * x * exp(x)) ).
%F A360466 E.g.f.: A(x) = -LambertW(-2 * x * exp(x)) / (2 * x * exp(x)).
%F A360466 E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (2 * x * exp(x))^k / k!.
%F A360466 a(n) = Sum_{k=0..n} 2^k * k^(n-k) * (k+1)^(k-1) * binomial(n,k).
%F A360466 a(n) ~ sqrt(1+LambertW(exp(-1)/2)) * n^(n-1) / (exp(n-1) * LambertW(exp(-1)/2)^n). - _Vaclav Kotesovec_, Feb 08 2023
%o A360466 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-2*x*exp(x)))))
%o A360466 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(-lambertw(-2*x*exp(x))/(2*x*exp(x))))
%o A360466 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(2*x*exp(x))^k/k!)))
%o A360466 (PARI) a(n) = sum(k=0, n, 2^k*k^(n-k)*(k+1)^(k-1)*binomial(n, k));
%Y A360466 Cf. A273954, A357247, A360465.
%K A360466 nonn
%O A360466 0,2
%A A360466 _Seiichi Manyama_, Feb 08 2023