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A361182 E.g.f. satisfies A(x) = exp( 3*x*A(x) ) / (1-x).

%I A361182 #20 Feb 16 2025 08:34:04
%S A361182 1,4,41,735,19293,672573,29342241,1540097541,94579646553,
%T A361182 6656561754345,528414534842949,46716837535074897,4552821617337191637,
%U A361182 484953672676323320109,56056228305888242732841,6988787950179969557086797,934866118278080385555647025
%N A361182 E.g.f. satisfies A(x) = exp( 3*x*A(x) ) / (1-x).
%H A361182 Seiichi Manyama, <a href="/A361182/b361182.txt">Table of n, a(n) for n = 0..329</a>
%H A361182 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A361182 a(n) = n! * Sum_{k=0..n} 3^k * (k+1)^(k-1) * binomial(n,k)/k!.
%F A361182 E.g.f.: LambertW( -3*x/(1-x) ) / (-3*x).
%F A361182 a(n) ~ (1 + 3*exp(1))^(n + 3/2) * n^(n-1) / (3^(3/2) * exp(n + 1/2)). - _Vaclav Kotesovec_, Mar 03 2023
%t A361182 nmax = 20; A[_] = 1;
%t A361182 Do[A[x_] = Exp[3*x*A[x]]/(1 - x) + O[x]^(nmax+1) // Normal, {nmax}];
%t A361182 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)
%o A361182 (PARI) a(n) = n!*sum(k=0, n, 3^k*(k+1)^(k-1)*binomial(n, k)/k!);
%o A361182 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(lambertw(-3*x/(1-x))/(-3*x)))
%Y A361182 Cf. A352410, A352448.
%Y A361182 Cf. A361066.
%K A361182 nonn
%O A361182 0,2
%A A361182 _Seiichi Manyama_, Mar 03 2023