A213720 a(0)=0, a(1)=0; for n>1, a(n) = a(n-1) + (n+2)*a(n-2) + 1.

0, 0, 1, 2, 9, 24, 97, 314, 1285, 4740, 20161, 81782, 364037, 1590768, 7415361, 34458418, 167934917, 822644860, 4181343201, 21456885262, 113446435685, 606954796712, 3329669253153, 18503539170954, 105074939752933, 604670497368692, 3546768810450817, 21082213234142886

Offset: 0

Author

Olivier Gérard, Nov 02 2012

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Programs

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

Mathematica

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 *)
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 *)
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 *)

Formula

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

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

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

Extensions

More terms from Vincenzo Librandi, Dec 24 2012

Edited by Bruno Berselli, Dec 24 2012