A323673 Expansion of e.g.f. log(1 - LambertW(-x)*(2 + LambertW(-x))/2).

0, 1, 0, 2, 7, 69, 696, 9400, 148506, 2753793, 58255840, 1388008566, 36768832200, 1072407094693, 34151921130432, 1179292944433500, 43892264744070736, 1751768399754149025, 74633720517351765504, 3380997879130123703818, 162286529338732345488000, 8227876237310253918100581

Offset: 0

Author

Ilya Gutkovskiy, Jan 23 2019

Keywords

• nonn

Crossrefs

Cf. A000272, A001858, A001865, A133297.

Programs

Maple

seq(n!*coeff(series(log(1-LambertW(-x)*(2+LambertW(-x))/2),x=0,22),x,n),n=0..21); # Paolo P. Lava, Jan 28 2019

Mathematica

nmax = 21; CoefficientList[Series[Log[1 - LambertW[-x] (2 + LambertW[-x])/2], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = n^(n - 2) - Sum[Binomial[n, k] (n - k)^(n - k - 2) k a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 21}]

Formula

E.g.f.: log(1 + Sum_{k>=1} k^(k-2)*x^k/k!).

a(0) = 0; a(n) = A000272(n) - (1/n)*Sum_{k=1..n-1} binomial(n,k)*A000272(n-k)*k*a(k).

a(n) ~ 2 * n^(n-2) / 3. - Vaclav Kotesovec, Jan 24 2019