A323772 - OEIS

%I #13 Jan 29 2019 05:02:40

%S 1,1,3,15,112,1165,15966,275149,5743032,141020793,3984082570,

%T 127298787121,4538547029556,178610366328277,7690287949961358,

%U 359592884584517445,18146340023779538416,982966789391874234865,56889414275458791370770,3503393307156206473624153,228738978280736413137020460

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

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

%F a(n) ~ (1 + exp(-1))^(n + 3/2) * n^(n-2). - _Vaclav Kotesovec_, Jan 27 2019

%p 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

%t nmax = 20; CoefficientList[Series[1 - LambertW[-x/(1 - x)] (2 + LambertW[-x/(1 - x)])/2, {x, 0, nmax}], x] Range[0, nmax]!

%t Join[{1}, Table[Sum[Binomial[n - 1, k - 1] k^(k - 2) n!/k!, {k, n}], {n, 20}]]

%Y Cf. A000272, A052871, A305276.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Jan 27 2019