A372321 Expansion of e.g.f. -exp( x + LambertW(-3*x)/3 ).

-1, 0, 6, 81, 1620, 45765, 1671678, 74794671, 3958829640, 241898775273, 16756621904970, 1297547591499819, 111065107263415308, 10412999996499836541, 1061234184094567585326, 116812280111404106348415, 13810631408232372091755792, 1745470697932523785587735249

Offset: 0

Author

Seiichi Manyama, Apr 27 2024

Keywords

• sign

Links

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

Crossrefs

Cf. A088957, A360193, A372315, A372316, A372320.

Programs

PARI

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

PARI

a(n) = sum(k=0, n, (3*k-1)^(k-1)*binomial(n, k));

Formula

a(n) = Sum_{k=0..n} (3*k-1)^(k-1) * binomial(n,k).

G.f.: Sum_{k>=0} (3*k-1)^(k-1) * x^k / (1-x)^(k+1).

a(n) ~ 3^(n-1) * n^(n-1) * exp((exp(-1) - 1)/3). - Vaclav Kotesovec, May 06 2024