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%I #27 Jul 14 2023 18:09:51
%S 1,1,2,5,12,36,113,351,1080,3281,9882,29646,88817,266085,797526,
%T 2391485,7173360,21520080,64563521,193700403,581120892,1743392201,
%U 5230206126,15690618378,47071766561,141215033961,423644570442,1270932914165,3812797945332,11438393835996
%N a(n) = n! * Sum_{i+2j+3k=n} 1/(i!*(2j)!*(3k)!).
%H G. C. Greubel, <a href="/A094717/b094717.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-12,10,-6,12,-9).
%F Limit_{n->oo} a(n)/3^n = 1/6.
%F E.g.f.: exp(z)*cosh(z)*(exp(z) + 2*exp(-z/2)*cos(z*sqrt(3/4)))/3. - _Peter Luschny_, Jul 11 2012
%F G.f.: (1-5*x+8*x^2-5*x^3+2*x^4-2*x^5)/((1-x)*(1-3*x)*(1+x+x^2)*(1-3*x+3*x^2)). - _Colin Barker_, Dec 24 2012
%F From _G. C. Greubel_, Jul 14 2023: (Start)
%F a(n) = (1/6)*(1 + 3^n + 2*A049347(n) + A049347(n-1) + 2*A057083(n) - 3*A057083(n-1)).
%F a(n) = (1/6)*(1 + 3^n + A099837(n+3) + A057682(n+3)). (End)
%p A094717_list := proc(n) local i; exp(z)*cosh(z)*(exp(z)+2*exp(-z/2)* cos(z*sqrt(3/4)))/3; series(%,z,n+2); seq(simplify(i!*coeff(%,z,i)),i=0..n) end: A094717_list(27); # _Peter Luschny_, Jul 11 2012
%t a[n_]:= n! Sum[Boole[i +2j +3k ==n]/(i! (2j)! (3k)!), {i,0,n}, {j,0,n}, {k,0,n}]; Table[a[n], {n,0,27}] (* _Jean-François Alcover_, Jul 06 2019 *)
%t LinearRecurrence[{6,-12,10,-6,12,-9}, {1,1,2,5,12,36}, 40] (* _G. C. Greubel_, Jul 14 2023 *)
%o (PARI) a(n)=sum(i=0,n,sum(j=0,n,sum(k=0,n,if(n-i-2*j-3*k,0,n!/(i)!/(2*j)!/(3*k)!))))
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-5*x+8*x^2-5*x^3+2*x^4-2*x^5)/((1-x)*(1-3*x)*(1+x+x^2)*(1-3*x+3*x^2)) )); // _G. C. Greubel_, Jul 14 2023
%o (SageMath)
%o def A094717_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( (1-5*x+8*x^2-5*x^3+2*x^4-2*x^5)/((1-x)*(1-3*x)*(1+x+x^2)*(1-3*x+3*x^2)) ).list()
%o A094717_list(40) # _G. C. Greubel_, Jul 14 2023
%Y Cf. A024493, A049347, A057083, A057682, A094715, A099837.
%K nonn
%O 0,3
%A _Benoit Cloitre_, May 23 2004
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