A362735 E.g.f. satisfies A(x) = exp(x + x / A(x)^2).

1, 2, -4, 56, -1008, 25632, -833600, 33067904, -1548418816, 83597525504, -5112566055936, 349330707068928, -26374805535322112, 2180554321981349888, -195926186031705505792, 19010400989418574020608, -1980997069982960384409600, 220651645970702249702326272

Offset: 0

Author

Seiichi Manyama, May 01 2023

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..339
• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Cf. A349562, A362693, A362694, A362734.

Cf. A349720.

Programs

PARI

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

Formula

E.g.f.: sqrt( 2*x / LambertW(2*x*exp(-2*x)) ) = exp( x + LambertW(2*x*exp(-2*x))/2 ).

a(n) = Sum_{k=0..n} (-2*k+1)^(n-1) * binomial(n,k) = 2^n * A349720(n).