A376101 Expansion of e.g.f. -LambertW(-x / (1 - 3*x)).

0, 1, 8, 99, 1684, 36865, 994986, 32106655, 1209994808, 52281293697, 2551380861070, 138903509144191, 8350198884092484, 549502839975044449, 39295464010757324930, 3034457861009541582015, 251666093876245502584816, 22310882229970705663827457

Offset: 0

Author

Seiichi Manyama, Sep 10 2024

Keywords

• nonn

Links

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

Crossrefs

Cf. A052871, A376100.

Programs

PARI

my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(-x/(1-3*x)))))

PARI

a(n) = n!*sum(k=1, n, 3^(n-k)*k^(k-1)*binomial(n-1, k-1)/k!);

Formula

E.g.f. A(x) satisfies A(x) = x * (3*A(x) + exp(A(x))).

E.g.f.: Series_Reversion( x / (3*x + exp(x)) ).

a(n) = n! * Sum_{k=1..n} 3^(n-k) * k^(k-1) * binomial(n-1,k-1)/k!.

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