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%I #35 Nov 28 2021 02:48:17
%S 1,1,4,37,532,10426,259300,7823908,277713904,11339452792,523621438336,
%T 26982030104536,1534947906550528,95550736737542464,
%U 6460746383585984512,471533064029919744256,36946948091091750496000,3093472887944746070621056
%N E.g.f. satisfies: A(x) = exp( x * (1 + A(x)^3)/2 ).
%H Seiichi Manyama, <a href="/A349714/b349714.txt">Table of n, a(n) for n = 0..352</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F a(n) = (1/2^n) * Sum_{k=0..n} (3*k+1)^(n-1) * binomial(n,k).
%F E.g.f.: ( -LambertW( -3*x/2 * exp(3*x/2) )/(3*x/2) )^(1/3).
%F G.f.: 2 * Sum_{k>=0} (3*k+1)^(k-1) * x^k/(2 - (3*k+1)*x)^(k+1).
%F a(n) ~ sqrt(1 + LambertW(exp(-1))) * 3^(n-1) * n^(n-1) / (LambertW(exp(-1))^(n + 1/3) * 2^n * exp(n)). - _Vaclav Kotesovec_, Nov 26 2021
%t a[n_] := (1/2^n) * Sum[(3*k + 1)^(n - 1) * Binomial[n, k], {k, 0, n}]; Array[a, 18, 0] (* _Amiram Eldar_, Nov 27 2021 *)
%o (PARI) a(n) = sum(k=0, n, (3*k+1)^(n-1)*binomial(n, k))/2^n;
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((-lambertw(-3*x/2*exp(3*x/2))/(3*x/2))^(1/3)))
%o (PARI) my(N=20, x='x+O('x^N)); Vec(2*sum(k=0, N, (3*k+1)^(k-1)*x^k/(2-(3*k+1)*x)^(k+1)))
%Y Cf. A007889, A202617, A349715, A349716, A349719, A349720, A349721.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Nov 26 2021
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