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%I #19 Sep 08 2022 08:45:55
%S 0,0,1,2,8,21,78,247,950,3421,13872,54925,235262,1004213,4533144,
%T 20600553,97664002,468473957,2324089996,11693569137,60499459054,
%U 317757980069,1709245538312,9335437059969,52066575517770,294787939076965,1700585478056756,9954647772211777
%N a(0)=0, a(1)=0; for n>1, a(n) = a(n-1) + (n+1)*a(n-2) + 1.
%H Vincenzo Librandi, <a href="/A185309/b185309.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = 2*a(n-1)+n*a(n-2)-n*a(n-3). - _Vincenzo Librandi_, Dec 24 2012
%F a(n) ~ (sqrt(Pi)/2 * (1 + (erf(1/sqrt(2))-1)*exp(1/2)) - 1/(2*sqrt(2))) * n^(n/2+1)*exp(sqrt(n)-n/2-1/4) * (1+31/(24*sqrt(n))). - _Vaclav Kotesovec_, Dec 26 2012
%F E.g.f.: 1/2*exp(-(x^2/2))*(sqrt(2*Pi)*erf(1/sqrt(2))*exp(x^2+x+1/2)*(x*(x+2)+2)+sqrt(2*Pi)*exp(x^2+x)*(x*(x+2)+2)*(erf(x/sqrt(2))-sqrt(exp(1))*erf((x+1)/sqrt(2)))-2*exp(x^2/2)*(x+1)-exp(x^2+x)*(x*(x+2)+2)+2*exp(1/2*x*(x+2))*(x+2)). - _Vaclav Kotesovec_, Dec 27 2012
%t RecurrenceTable[{a[1] == 0, a[2] == 0, a[n] == a[n - 1] + n a[n - 2] + 1}, a, {n, 30}] (* _Bruno Berselli_, Dec 24 2012 *)
%t FullSimplify[CoefficientList[Series[1/2*E^(-(x^2/2))*(Sqrt[2*Pi]*Erf[1/Sqrt[2]]*E^(x^2+x+1/2)*(x*(x+2)+2)+Sqrt[2*Pi]*E^(x^2+x)*(x*(x+2)+2)*(Erf[x/Sqrt[2]]-Sqrt[E]*Erf[(x+1)/Sqrt[2]])-2*E^(x^2/2)*(x+1)-E^(x^2+x)*(x*(x+2)+2)+2*E^(1/2*x*(x+2))*(x+2)), {x, 0, 20}], x]* Range[0, 20]!] (* _Vaclav Kotesovec_, Dec 27 2012 *)
%o (Magma) I:=[0,0,1,2]; [n le 4 select I[n] else 2*Self(n-1)+(n-1)*Self(n-2)-(n-1)*Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Dec 24 2012
%K nonn
%O 0,4
%A _Olivier Gérard_, Nov 02 2012
%E More terms from _Vincenzo Librandi_, Dec 24 2012
%E Edited by _Bruno Berselli_, Dec 24 2012
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