A185309 a(0)=0, a(1)=0; for n>1, a(n) = a(n-1) + (n+1)*a(n-2) + 1.

0, 0, 1, 2, 8, 21, 78, 247, 950, 3421, 13872, 54925, 235262, 1004213, 4533144, 20600553, 97664002, 468473957, 2324089996, 11693569137, 60499459054, 317757980069, 1709245538312, 9335437059969, 52066575517770, 294787939076965, 1700585478056756, 9954647772211777

Offset

0,4

Links

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

Formula

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

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

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

Mathematica

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 *)
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 *)

Prog

(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

Crossrefs

Sequence in context: A316270 A219970 A107361 * A318040 A333705 A085077

Adjacent sequences: A185306 A185307 A185308 * A185310 A185311 A185312

Keyword

nonn

Author

Olivier Gérard, Nov 02 2012

Extensions

More terms from Vincenzo Librandi, Dec 24 2012

Edited by Bruno Berselli, Dec 24 2012

Status

approved