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%I #20 Mar 22 2016 06:25:46
%S 0,2,3,10,35,191,1162,8996,77877,786757,8801276,110180038,1508049127,
%T 22568091079,364984569510,6360525167496,118634584548905,
%U 2360362530705801,49871009321693920,1115567129580176010,26332809559025886651
%N a(n) = (n+1)!*Sum_{k=0..(n-1)/2}((k)!*stirling2(n-k,k+1)/(n-k)!/(k+1)).
%F E.g.f.: -(((x+1)*e^x-1)*log(-x*e^x+x+1))/(x*e^x-x).
%F a(n) ~ (n-1)! * (1 + 1/r + r) / r^n, where r = 0.8064659942363268087699282186454... is the root of the equation exp(r) = 1 + 1/r. - _Vaclav Kotesovec_, Mar 22 2016
%t Table[(n+1)! * Sum[k!*StirlingS2[n-k, k+1]/(n-k)!/(k+1), {k, 0, (n-1)/2}], {n, 0, 20}] (* _Vaclav Kotesovec_, Mar 22 2016 *)
%o (Maxima)
%o makelist((n)!*coeff(taylor(-(((x+1)*%e^x-1)*log(-x*%e^x+x+1))/(x*%e^x-x),x,0,15),x,n),n,0,15);
%o a(n):=(n+1)!*sum((k)!*stirling2(n-k,k+1)/(n-k)!/(k+1),k,0,(n-1)/2);
%Y Cf. A048993.
%K nonn
%O 0,2
%A _Vladimir Kruchinin_, Mar 22 2016
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