A328008 - OEIS

%I #9 Oct 02 2019 06:21:35

%S 1,2,13,124,1575,25006,476421,10589720,269010979,7687905826,

%T 244120131393,8526912775756,324914136199263,13412430958497494,

%U 596253684006657085,28399969571266895488,1442890578572155475355,77889310498718258171914,4451905168738601015593785

%N Expansion of e.g.f. 1 / (2 - exp(x) / (1 - x)).

%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * A000522(k) * a(n-k).

%F a(n) ~ n! / (2*(1 + 1/LambertW(exp(1)/2)) * (1 - LambertW(exp(1)/2))^(n+1)). - _Vaclav Kotesovec_, Oct 02 2019

%t nmax = 18; CoefficientList[Series[1/(2 - Exp[x]/(1 - x)), {x, 0, nmax}], x] Range[0, nmax]!

%t a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Floor[Exp[1] k!] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]

%o (PARI) my(x='x+O('x^25)); Vec(serlaplace(1/(2-exp(x)/(1-x)))) \\ _Michel Marcus_, Oct 02 2019

%Y Cf. A000522, A002866, A321974, A328007.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Oct 01 2019