A361068 E.g.f. satisfies A(x) = exp( x/((1-x) * A(x)^2) ).

1, 1, -1, 13, -127, 2101, -41801, 1030177, -29820127, 995977801, -37660751569, 1590847310581, -74242656468575, 3793664894534269, -210656932372422745, 12630986901470435401, -813335155262348743231, 55977540398642247218449

Offset: 0

Author

Seiichi Manyama, Mar 01 2023

Keywords

• sign

Links

• Winston de Greef, Table of n, a(n) for n = 0..360
• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Cf. A052868, A361065, A361066, A361067, A361069.

Programs

Mathematica

nmax = 20; A[_] = 1;
Do[A[x_] = Exp[x/((1 - x)*A[x]^2)] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

PARI

a(n) = n!*sum(k=0, n, (-2*k+1)^(k-1)*binomial(n-1, n-k)/k!);

PARI

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

PARI

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

Formula

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

E.g.f.: exp( LambertW(2*x/(1-x))/2 ).

E.g.f.: 1 / sqrt( (1-x)/(2*x) * LambertW(2*x/(1-x)) ).

a(n) ~ (-1)^(n+1) * 2^(-3/2) * exp(-1/2) * (2 - exp(-1))^(n + 1/2) * n^(n-1). - Vaclav Kotesovec, Apr 22 2024