A220700 a(0)=0, a(1)=0; for n>1, a(n) = a(n-1) + (n+3)*a(n-2) + 1

0, 0, 1, 2, 10, 27, 118, 389, 1688, 6357, 28302, 117301, 541832, 2418649, 11629794, 55165477, 276131564, 1379441105, 7178203950, 37525908261, 202624599112, 1103246397377, 6168861375178, 34853267706981, 201412524836788, 1177304020632257, 7018267240899110

Offset

0,4

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Formula

a(0)=a(1)=0, a(2)=1, a(n) = 2*a(n-1)+(n+2)*a(n-2)-(n+2)*a(n-3).

E.g.f.: 1/8*exp(-(x^2/2))*(exp(x^2/2)*(3*sqrt(2*Pi)*erf(1/sqrt(2))*exp(1/2*(x+1)^2)*(x*(x+2)*(x*(x+2)+8)+10)-6*(x+1)*(x*(x+2)+6)-6*exp(1/2*x*(x+2))*(x*(x+2)*(x*(x+2)+8)+10)+8*exp(x)*(x*(x*(x+4)+11)+12))+sqrt(2*Pi)*exp(x^2+x)*(x*(x+2)*(x*(x+2)+8)+10)*(4*erf(x/sqrt(2))-3*sqrt(exp(1))*erf((x+1)/sqrt(2)))). - Vaclav Kotesovec, Dec 27 2012

a(n) ~ (1/2*sqrt(Pi)-3/(4*sqrt(2))+3/8*sqrt(Pi)*exp(1/2)*(erf(1/sqrt(2))-1)) * n^(n/2+2)*exp(sqrt(n)-n/2-1/4) * (1+55/(24*sqrt(n))). - Vaclav Kotesovec, Dec 27 2012

Mathematica

RecurrenceTable[{a[0] == 0, a[1] == 0, a[n] == a[n-1] + (n+3) a[n-2] + 1}, a, {n, 0, 40}] (* corrected by Georg Fischer, Dec 05 2019 *)
FullSimplify[CoefficientList[Series[1/8*E^(-(x^2/2))*(E^(x^2/2)*(3*Sqrt[2*Pi]*Erf[1/Sqrt[2]]*E^(1/2*(x+1)^2)*(x*(x+2)*(x*(x+2)+8)+10)-6*(x+1)*(x*(x+2)+6)-6*E^(1/2*x*(x+2))*(x*(x+2)*(x*(x+2)+8)+10)+8*E^x*(x*(x*(x+4)+11)+12))+Sqrt[2*Pi]*E^(x^2+x)*(x*(x+2)*(x*(x+2)+8)+10)*(4*Erf[x/Sqrt[2]]-3*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+4)+1}; NestList[nxt, {1, 0, 0}, 30][[All, 2]] (* Harvey P. Dale, Mar 01 2020 *)

Prog

(Magma) [n le 2 select 0 else Self(n-1)+(n+2)*Self(n-2) + 1: n in [1..30]];

Crossrefs

Cf. A185108, A185109, A185308, A185309, A186738, A186739, A213720.

Sequence in context: A196324 A327841 A196648 * A057753 A060515 A109723

Adjacent sequences: A220697 A220698 A220699 * A220701 A220702 A220703

Keyword

nonn,easy

Author

Vincenzo Librandi, Dec 25 2012

Status

approved