A372155 E.g.f. A(x) satisfies A(x) = exp( 3 * x * (1 + x) * A(x)^(1/3) ).

1, 3, 21, 198, 2505, 39348, 743967, 16465494, 418281393, 12006610344, 384595471119, 13607063765298, 527217367699881, 22209587195328588, 1010947593782034687, 49457001919808733102, 2588247541696766293857, 144302243002459116148944

Offset: 0

Author

Seiichi Manyama, Apr 20 2024

Keywords

• nonn

Links

• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Cf. A362771, A372154.

Programs

PARI

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

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(s*k, n-k)/k!);

Formula

E.g.f.: A(x) = exp( -3 * LambertW(-x * (1+x)) ).

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(s*k,n-k)/k!.