%I #21 Mar 04 2024 08:04:56

%S 1,1,1,4,9,76,175,3606,833,354376,-1605249,65111410,-718371071,

%T 20105327100,-351241054177,9362931464446,-214514949732735,

%U 6039303900168976,-165679758877120001,5093296357218337386,-159900268661169533119,5405435526807425433220

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

%H Winston de Greef, <a href="/A361067/b361067.txt">Table of n, a(n) for n = 0..413</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} (-k+1)^(k-1) * binomial(n-1,n-k)/k!.

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

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

%F a(n) ~ -(-1)^n * exp(-1) * (1 - exp(-1))^(n + 1/2) * n^(n-1). - _Vaclav Kotesovec_, Mar 02 2023

%t nmax = 21; A[_] = 1;

%t Do[A[x_] = Exp[x/((1 - x)*A[x])] + O[x]^(nmax+1) // Normal, {nmax}];

%t CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)

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

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(x/(1-x)))))

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/((1-x)/x*lambertw(x/(1-x)))))

%Y Cf. A052868, A361065, A361066, A361068, A361069.

%K sign

%O 0,4

%A _Seiichi Manyama_, Mar 01 2023