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%I #13 Jun 02 2013 10:40:02
%S 0,0,1,2,11,30,141,472,2165,8302,38613,163144,780953,3554402,17611557,
%T 85145196,437376337,2225425454,11847704869,63032490312,347377407169,
%U 1923189664970,10955002251365,62881123205556,369621186243777,2193173759204902,13281809346518213
%N 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.
%C 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.
%C 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]
%H Vincenzo Librandi, <a href="/A187830/b187830.txt">Table of n, a(n) for n = 0..300</a>
%F 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))
%F 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))))
%t 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}]
%t 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]!]
%Y Cf. A220700 (k=2), A213720 (k=1), A185309 (k=0), A185308 (k=-1), A186738 (k=-2), A186739 (k=-3), A193361 (k=-4), A220699 (k=-5).
%K nonn
%O 0,4
%A _Vaclav Kotesovec_, Dec 27 2012
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