A376098 Expansion of e.g.f. -LambertW(-2*x / (1 - x))/2.

0, 1, 6, 66, 1112, 25640, 753552, 26950000, 1136106624, 55167345792, 3032389917440, 186130732899584, 12619351769121792, 936591263680543744, 75527892444435486720, 6575887645386829301760, 614790327790529665138688, 61429094739085165675446272

Offset: 0

Author

Seiichi Manyama, Sep 10 2024

Keywords

• nonn

Links

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

Crossrefs

Cf. A052871, A376099.

Cf. A352448, A376093.

Programs

PARI

my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(-2*x/(1-x))/2)))

PARI

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

Formula

E.g.f. A(x) satisfies A(x) = x * (A(x) + exp(2*A(x))).

E.g.f.: Series_Reversion( x / (x + exp(2*x)) ).

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

a(n) = n * A352448(n-1).

a(n) ~ (2 + exp(-1))^(n + 1/2) * n^(n-1) / 2^(3/2). - Vaclav Kotesovec, Sep 10 2024