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E.g.f. satisfies A(x) = exp(x + x * A(x)^3).
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%I #23 Feb 16 2025 08:34:05

%S 1,2,16,296,8512,333632,16595200,1001460224,71094759424,5805799829504,

%T 536188352856064,55259197654089728,6287146625230962688,

%U 782751635353947865088,105852868748672770244608,15451195442132410179780608,2421355190097788960505856000

%N E.g.f. satisfies A(x) = exp(x + x * A(x)^3).

%H Seiichi Manyama, <a href="/A362734/b362734.txt">Table of n, a(n) for n = 0..322</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.

%F E.g.f.: ( -LambertW(-3*x*exp(3*x)) / (3*x) )^(1/3) = exp( x - LambertW(-3*x*exp(3*x))/3 ).

%F a(n) = Sum_{k=0..n} (3*k+1)^(n-1) * binomial(n,k) = 2^n * A349714(n).

%F a(n) ~ sqrt(LambertW(exp(-1)) + 1) * 3^(n-1) * n^(n-1) / (exp(n) * LambertW(exp(-1))^(n + 1/3)). - _Vaclav Kotesovec_, Apr 24 2024

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-3*x*exp(3*x))/3)))

%Y Cf. A349562, A362693, A362694, A362735.

%Y Cf. A349714, A362392, A362472.

%K nonn,changed

%O 0,2

%A _Seiichi Manyama_, May 01 2023