A213720 - OEIS

%I #32 Dec 08 2024 10:19:04

%S 0,0,1,2,9,24,97,314,1285,4740,20161,81782,364037,1590768,7415361,

%T 34458418,167934917,822644860,4181343201,21456885262,113446435685,

%U 606954796712,3329669253153,18503539170954,105074939752933,604670497368692,3546768810450817,21082213234142886

%N a(0)=0, a(1)=0; for n>1, a(n) = a(n-1) + (n+2)*a(n-2) + 1.

%H Vincenzo Librandi, <a href="/A213720/b213720.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = 2*a(n-1)+(n+1)*a(n-2)-(n+1)*a(n-3) with a(0)=a(1)=0, a(2)=1. - _Vincenzo Librandi_, Dec 24 2012

%F E.g.f.: 1/6*exp(-(x^2/2))*(exp(x^2/2)*(sqrt(2*Pi)*erf(1/sqrt(2))*exp(1/2*(x+1)^2)*(x+1)*(x*(x+2)+4)-2*(x*(x+2)+3)-6*exp(1/2*x*(x+2))*(x+1)*(x*(x+2)+4)+6*exp(x)*(x*(x+3)+5))+sqrt(2*Pi)*exp(x^2+x)*(x+1)*(x*(x+2)+4)*(3*erf(x/sqrt(2))-sqrt(exp(1))*erf((x+1)/sqrt(2)))). - _Vaclav Kotesovec_, Dec 27 2012

%F a(n) ~ (1/2*sqrt(Pi)-1/sqrt(2)+1/6*sqrt(Pi)*exp(1/2)*(erf(1/sqrt(2))-1)) * n^(n/2+3/2)*exp(sqrt(n)-n/2-1/4) * (1+43/(24*sqrt(n))). - _Vaclav Kotesovec_, Dec 27 2012

%t RecurrenceTable[{a[1] == 0, a[2] == 0, a[n] == a[n - 1] + (n + 1) a[n - 2] + 1}, a, {n, 30}] (* _Bruno Berselli_, Dec 24 2012 *)

%t FullSimplify[CoefficientList[Series[1/6*E^(-(x^2/2))*(E^(x^2/2)*(Sqrt[2*Pi]*Erf[1/Sqrt[2]]*E^(1/2*(x+1)^2)*(x+1)*(x*(x+2)+4)-2*(x*(x+2)+3)-6*E^(1/2*x*(x+2))*(x+1)*(x*(x+2)+4)+6*E^x*(x*(x+3)+5))+Sqrt[2*Pi]*E^(x^2+x)*(x+1)*(x*(x+2)+4)*(3*Erf[x/Sqrt[2]]-Sqrt[E]*Erf[(x+1)/Sqrt[2]])), {x, 0, 20}], x]* Range[0, 20]!] (* _Vaclav Kotesovec_, Dec 27 2012 *)

%t nxt[{n_,a_,b_}]:={n+1,b,b+a(n+3)+1}; NestList[nxt,{1,0,0},30][[;;,2]] (* _Harvey P. Dale_, Nov 23 2024 *)

%o (Magma) I:=[0,0,1,2]; [n le 4 select I[n] else 2*Self(n-1)+n*Self(n-2)-n*Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Dec 24 2012

%K nonn,changed

%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