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E.g.f. A(x) satisfies A(x) = exp( 3 * x * A(x)^(1/3) / (1 - x) ).
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%I #14 Apr 21 2024 11:48:58

%S 1,3,21,216,2937,49788,1013247,24106134,657277185,20225122632,

%T 693755934159,26261393088978,1087866116802081,48965716033901436,

%U 2380245527593532559,124300353332797939422,6941285402232405794817,412817223292008085699344

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

%o (PARI) a(n, r=3, 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, A372158.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Apr 20 2024