A372156 E.g.f. A(x) satisfies A(x) = exp( 2 * x * (1 + x * A(x)^(1/2)) ).

1, 2, 8, 44, 328, 3032, 33964, 445580, 6727984, 114892784, 2192201044, 46233324788, 1068561369352, 26865052934696, 730137962157244, 21334636036296668, 667074635111434336, 22225983296836137440, 786215841115748129956, 29429693502599243538884

Offset: 0

Author

Seiichi Manyama, Apr 20 2024

Keywords

• nonn

Links

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

Crossrefs

Cf. A125500, A372157.

Programs

PARI

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

PARI

a(n, r=2, s=1, t=0, u=1) = 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*x - 2*LambertW(-x^2 * exp(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(1 + LambertW(exp(-1/2)/2)) * n^(n-1) / (LambertW(exp(-1/2)/2)^(n+4) * 2^(n + 5/2) * exp(n)). - Vaclav Kotesovec, Aug 05 2025