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A187830
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a(n)=2*a(n-1)+(n+3)*a(n-2)-(n+3)*a(n-3), a(0)=0, a(1)=0, a(2)=1.
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2
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0, 0, 1, 2, 11, 30, 141, 472, 2165, 8302, 38613, 163144, 780953, 3554402, 17611557, 85145196, 437376337, 2225425454, 11847704869, 63032490312, 347377407169, 1923189664970, 10955002251365, 62881123205556, 369621186243777, 2193173759204902, 13281809346518213
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OFFSET
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0,4
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COMMENTS
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This is case k=3. In general case, recurrence a(n)=2*a(n-1)+(n+k)*(a(n-2)-a(n-3)) is asymptotic to a(n) ~ c * n^(n/2+k/2+1)*exp(sqrt(n)-n/2-1/4) * (1+(12*k+31)/(24*sqrt(n))), where c is constant dependent only on k.
EGF is solution of the equation DSolve[{(3+k)*f[x] + (x-3-k)*f'[x] - (x+2)*f''[x] + f'''[x]==0, f[0]==0, f'[0]==0, f''[0]==1}, f, x]
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LINKS
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FORMULA
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E.g.f.: 1/30*exp(-(x^2/2))*((8*sqrt(2*exp(1)*Pi)*erf(1/sqrt(2))-27)*exp(x^2+x)*(x+1)*(x*(x+2)*(x*(x+2)+12)+26)+sqrt(2*Pi)*exp(x^2+x)*(x+1)*(x*(x+2)*(x*(x+2)+12)+26)*(15*erf(x/sqrt(2))-8*sqrt(exp(1))*erf((x+1)/sqrt(2)))-16*exp(x^2/2)*(x*(x+2)+2)*(x*(x+2)+9)+30*exp(1/2*x*(x+2))*(x*(x*(x*(x+5)+19)+35)+33))
a(n) ~ (1/2*sqrt(Pi)-9/(10*sqrt(2))+4/15*sqrt(Pi)*exp(1/2)*(erf(1/sqrt(2))-1)) * n^(n/2+5/2)*exp(sqrt(n)-n/2-1/4) * (1+(67/(24*sqrt(n))))
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MATHEMATICA
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RecurrenceTable[{(3+n)*a[-3+n]+(-3-n)*a[-2+n]-2*a[-1+n]+a[n]==0, a[0]==0, a[1]==0, a[2]==1}, a, {n, 20}]
FullSimplify[CoefficientList[Series[1/30*E^(-(x^2/2))*((8*Sqrt[2*E*Pi]*Erf[1/Sqrt[2]]-27)*E^(x^2+x)*(x+1)*(x*(x+2)*(x*(x+2)+12)+26)+Sqrt[2*Pi]*E^(x^2+x)*(x+1)*(x*(x+2)*(x*(x+2)+12)+26)*(15*Erf[x/Sqrt[2]]-8*Sqrt[E]*Erf[(x+1)/Sqrt[2]])-16*E^(x^2/2)*(x*(x+2)+2)*(x*(x+2)+9)+30*E^(1/2*x*(x+2))*(x*(x*(x*(x+5)+19)+35)+33)), {x, 0, 20}], x]* Range[0, 20]!]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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