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E.g.f. satisfies: A(x) = exp( x * (1 + 1/A(x)^3)/2 ).
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%I #21 Dec 05 2021 05:32:07

%S 1,1,-2,19,-260,4966,-121328,3613996,-127035920,5147600680,

%T -236245559984,12112405259560,-686148484748480,42560312499982720,

%U -2868921992458611200,208828244778853125376,-16324500711130356582656,1363986660232205656646272

%N E.g.f. satisfies: A(x) = exp( x * (1 + 1/A(x)^3)/2 ).

%H Seiichi Manyama, <a href="/A349721/b349721.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.: ( (3*x/2)/LambertW( 3*x/2 * exp(-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) ~ -(-1)^n * sqrt(1 + LambertW(exp(-1))) * 3^(n-1) * n^(n-1) / (2^n * exp(n) * LambertW(exp(-1))^(n - 1/3)). - _Vaclav Kotesovec_, Dec 05 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(((3*x/2)/lambertw(3*x/2*exp(-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, A349714, A349715, A349716, A349719, A349720.

%K sign

%O 0,3

%A _Seiichi Manyama_, Nov 27 2021