%I #22 May 25 2024 21:11:39

%S 1,3,21,288,5841,158148,5370003,219641922,10518990129,577629889848,

%T 35788733371179,2470154920005798,187970878034549001,

%U 15636177199793409444,1411635193678825868979,137469669176542404342042,14364540773583252035937633

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

%H Winston de Greef, <a href="/A361214/b361214.txt">Table of n, a(n) for n = 0..338</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)^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)).

%t Table[(-1)^n*n!*Sum[(-3)^k*(k + 1)^(k - 1)*Binomial[n - 1, n - k]/k!, {k, 0, n}], {n, 0, 20}] (* _Wesley Ivan Hurt_, May 25 2024 *)

%o (PARI) a(n) = (-1)^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. A335945, A361213.

%Y Cf. A361069, A361194.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Mar 04 2023