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

%I #38 Mar 09 2024 08:15:55

%S 1,2,17,313,9053,357941,17975605,1095604133,78570635225,6482415935449,

%T 604889610870881,62989604872166897,7241672622495518773,

%U 911048848278644776949,124497704904842673086285,18364053909500922198147421,2908158473059042016441887025

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

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

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

%F a(n) ~ 3^(-5/6) * (2^(4/3) + 2*(3 + sqrt(4*exp(1) + 9))^(1/3) * exp(-2/3) - 2^(2/3) * (3 + sqrt(4*exp(1) + 9))^(2/3) * exp(-1/3))^(1/6) * 2^(1/3) * (3 + sqrt(4*exp(1) + 9))^(4/9) * sqrt(4 - 2^(4/3) * (3 + sqrt(4*exp(1) + 9))^(2/3) * exp(-1/3) + 3*2^(2/3) * exp(-2/3) * (3 + sqrt(4*exp(1) + 9))^(1/3)) * n^(n-1) * (12 + 4*sqrt(4*exp(1) + 9))^(n/3) / (exp(7/18 + 5*n/3) * (2 - 2^(1/3) * (3 + sqrt(4*exp(1) + 9))^(2/3) * exp(-1/3) + exp(-2/3) * (12 + 4*sqrt(4*exp(1) + 9))^(1/3))^n * ((3 + sqrt(4*exp(1) + 9))^(2/3) * exp(-1/3) - 2^(2/3))^(3/2) * sqrt(2^(1/3) * (3 + sqrt(4*exp(1) + 9))^(2/3) * exp(-1/3) - 2)). - _Vaclav Kotesovec_, Mar 06 2023

%F a(n) = n! * Sum_{k=0..n} (3*k+1)^(k-1) * binomial(n+2*k,n-k)/k!. - _Seiichi Manyama_, Mar 09 2024

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((lambertw(-3*x/(1-x)^3)/(-3*x))^(1/3)))

%Y Cf. A352410, A360601.

%Y Cf. A052752, A361182.

%Y Cf. A370876.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Mar 05 2023