A396680 Expansion of e.g.f. -log(1 + W_3(-x)), where W_k(x) is the k-th iterate of LambertW(x).

1, 7, 83, 1386, 29889, 790878, 24820221, 901692088, 37234618065, 1723051928970, 88341418678149, 4971522602635020, 304711715522655609, 20206757357179068718, 1441624953145861003725, 110109081333313185170832, 8964827662941801614166561, 775105294977887655831830802

Offset

1,2

Links

Table of n, a(n) for n=1..18.

Formula

a(n) = Sum_{k=1..n} n^(n-k) * binomial(n-1,k-1) * A396679(k).

a(n) = (n-1)! * Sum_{i,j,k,l >= 0 and i+j+k+l=n-1} n^i * (n-i)^j * (n-i-j)^k / (i!*j!*k!).

a(n) ~ sqrt(Pi/2) * n^(n - 1/2) * exp(n*(exp(-1) + exp(-1-exp(-1)))) * (1 - exp(1 + exp(-1)/2) / (3*sqrt((exp(1)-1) * (exp(1 + 1/exp(1))-1)*Pi*n/2))). - Vaclav Kotesovec, Jun 29 2026

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(-log(1+lambertw(lambertw(lambertw(-x))))))

Crossrefs

Column k=3 of A396676.

Cf. A227278, A396678, A396679.

Sequence in context: A386920 A388727 A231259 * A383231 A187649 A368853

Adjacent sequences: A396677 A396678 A396679 * A396682 A396683 A396684

Keyword

nonn

Author

Seiichi Manyama, Jun 02 2026

Status

approved