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E.g.f. satisfies A(x) = exp( 3*x*A(x) / (1-x) ).
2

%I #18 Mar 12 2023 15:32:09

%S 1,3,33,612,16353,576108,25306803,1334701854,82258866225,

%T 5805344935368,461848917299499,40904277651802458,3992219566916292873,

%U 425766991650939828828,49266876888419716251315,6147944525591645916094182,823045511075200872642258273

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

%H Winston de Greef, <a href="/A361212/b361212.txt">Table of n, a(n) for n = 0..327</a>

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

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

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

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

%o (PARI) a(n) = n!*sum(k=0, n, 3^k*(k+1)^(k-1)*binomial(n-1, n-k)/k!);

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

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

%Y Cf. A052868, A360939.

%Y Cf. A361066, A361182.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Mar 04 2023