A372154 E.g.f. A(x) satisfies A(x) = exp( 2 * x * (1 + x) * A(x)^(1/2) ).

1, 2, 12, 98, 1128, 16442, 293356, 6195114, 151432112, 4209004466, 131188519764, 4533821784098, 172125130420744, 7122734349079338, 319148172778019708, 15395906192167996058, 795673541794111734624, 43862837291529529270370

Offset: 0

Author

Seiichi Manyama, Apr 20 2024

Keywords

• nonn

Links

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

Crossrefs

Cf. A362771, A372155.

Programs

PARI

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

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

Formula

E.g.f.: A(x) = exp( -2 * 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!.

a(n) ~ sqrt(2 + 8*exp(-1) - 2*sqrt(1 + 4*exp(-1))) * 2^n * n^(n-1) / ((sqrt(1+4*exp(-1)) - 1)^n * exp(n - 5/2)). - Vaclav Kotesovec, Aug 05 2025