A323772 Expansion of e.g.f. 1 - LambertW(-x/(1 - x))*(2 + LambertW(-x/(1 - x)))/2.

1, 1, 3, 15, 112, 1165, 15966, 275149, 5743032, 141020793, 3984082570, 127298787121, 4538547029556, 178610366328277, 7690287949961358, 359592884584517445, 18146340023779538416, 982966789391874234865, 56889414275458791370770, 3503393307156206473624153, 228738978280736413137020460

Offset: 0

Author

Ilya Gutkovskiy, Jan 27 2019

Keywords

• nonn

Crossrefs

Cf. A000272, A052871, A305276.

Programs

Maple

seq(n!*coeff(series(1-LambertW(-x/(1-x))*(2+LambertW(-x/(1-x)))/2,x=0,21),x,n),n=0..20); # Paolo P. Lava, Jan 29 2019

Mathematica

nmax = 20; CoefficientList[Series[1 - LambertW[-x/(1 - x)] (2 + LambertW[-x/(1 - x)])/2, {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[Sum[Binomial[n - 1, k - 1] k^(k - 2) n!/k!, {k, n}], {n, 20}]]

Formula

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

a(n) ~ (1 + exp(-1))^(n + 3/2) * n^(n-2). - Vaclav Kotesovec, Jan 27 2019