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A220700
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a(0)=0, a(1)=0; for n>1, a(n) = a(n-1) + (n+3)*a(n-2) + 1
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2
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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
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OFFSET
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0,4
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LINKS
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(Magma) [n le 2 select 0 else Self(n-1)+(n+2)*Self(n-2) + 1: n in [1..30]];
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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