A372316 Expansion of e.g.f. exp( x - LambertW(-3*x)/3 ).

1, 2, 10, 125, 2644, 77597, 2904382, 132169403, 7083715240, 437031850841, 30506442905194, 2377038378159359, 204521399708464252, 19259006462435865413, 1970114326513629358654, 217556451608123850352523, 25794252755430105917806288, 3268152272130255473300883377

Offset: 0

Author

Seiichi Manyama, Apr 27 2024

Keywords

• nonn

Links

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

Crossrefs

Cf. A088957, A360193, A372315, A372320, A372321.

Cf. A362734.

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 04 2024