%I #13 Apr 21 2024 11:49:02

%S 1,2,12,110,1368,21602,415036,9416094,246730448,7340456258,

%T 244615296564,9030708939518,365998814372824,16159576541122146,

%U 772216069907880812,39715949460883093598,2187682975276318552224,128508919233259720967810

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

%F If e.g.f. satisfies A(x) = exp( r*x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s ), then a(n) = r * n! * Sum_{k=0..n} (t*k+u*(n-k)+r)^(k-1) * binomial(n+(s-1)*k-1,n-k)/k!.

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-2*lambertw(-x/(1-x)))))

%o (PARI) a(n, r=2, s=1, t=1, u=0) = r*n!*sum(k=0, n, (t*k+u*(n-k)+r)^(k-1)*binomial(n+(s-1)*k-1, n-k)/k!);

%Y Cf. A052868, A372159.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Apr 20 2024