A185308 a(0)=0, a(1)=0; for n>1, a(n) = a(n-1) + n*a(n-2) + 1.

0, 0, 1, 2, 7, 18, 61, 188, 677, 2370, 9141, 35212, 144905, 602662, 2631333, 11671264, 53772593, 252184082, 1220090757, 6011588316, 30413403457, 156656758094, 825751634149, 4428857070312, 24246896289889, 134968323047690, 765387626584805, 4409532348872436

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-2)*Self(n-2)-(n-2)*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Dec 24 2012

Mathematica

RecurrenceTable[{a[0] == 0, a[1] == 0, a[n] == a[n - 1] + n a[n - 2] + 1}, a, {n, 0, 30}] (* Bruno Berselli, Dec 24 2012 *)
FullSimplify[CoefficientList[Series[1/2*E^x*(E^(x^2/2)*(x+1)*(Sqrt[2*Pi]*Erf[x/Sqrt[2]]-2)+2), {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Dec 27 2012 *)

Formula

a(0)=a(1)=0, a(2)=1, a(n) = 2*a(n-1)+(n-1)*a(n-2)-(n-1)*a(n-3). - Vincenzo Librandi, Dec 24 2012

a(n) ~ (sqrt(Pi)-sqrt(2))/2 * n^(n/2+1/2)*exp(sqrt(n)-n/2-1/4) * (1+19/(24*sqrt(n))). - Vaclav Kotesovec, Dec 26 2012

E.g.f.: 1/2*exp(x)*(exp(x^2/2)*(x+1)*(sqrt(2*Pi)*erf(x/sqrt(2))-2)+2). - Vaclav Kotesovec, Dec 27 2012

Extensions

More terms from Vincenzo Librandi, Dec 24 2012

Edited by Bruno Berselli, Dec 24 2012