%I #33 Aug 02 2022 05:50:59
%S 1,1,2,2,4,8,12,28,60,112,284,652,1404,3776,9228,22028,62092,160448,
%T 414540,1216012,3302604,9092272,27622844,78446956,227652828,713772368,
%U 2110379772,6405093068,20668461340,63385346912,200011067244
%N a(n) = n!*b(n) where b(n) = (b(n-2) + b(n-3))/(n*(n-1)), b(0) = b(1) = b(2) = 1.
%D Richard Bronson, Schaum's Outline of Modern Introductory Differential Equations, MacGraw-Hill, New York,1973, page 107, solved problem 19.15.
%H G. C. Greubel, <a href="/A123024/b123024.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = n!*b(n) where b(n) = (b(n-2) + b(n-3))/(n*(n-1)), b(0) = b(1) = b(2) = 1.
%F From _G. C. Greubel_, Jul 10 2021: (Start)
%F D-finite with recurrence a(n) = a(n-2) + (n-2)*a(n-3), with a(0) = a(1) = 1, and a(2) = 2. [corrected by _Georg Fischer_, Jul 30 2022]
%F (End)
%F From _Vaclav Kotesovec_, Jul 31 2022: (Start)
%F E.g.f. A(x) satisfies the differential equation A'''(x) = A(x) + (x+1)*A'(x) with A(0) = 1, A'(0) = 1, A''(0) = 1.
%F a(n) ~ c * exp(n^(1/3) - n/3) * n^(n/3 - 1/6) * (1 + 1/(6*n^(1/3)) + 1/(72*n^(2/3)) - 107/(1296*n)), where c = 0.56651588691862348716469232164213071696766708621590... (End)
%t b[n_]:= b[n]= If[n<3, 1, (b[n-2] +b[n-3])/(n*(n-1))]; a[n_]:= n!*b[n]; Table[a[n], {n, 0, 30}]
%t RecurrenceTable[{a[0] == 1, a[1] == 1, a[2] == 2, a[n] == a[n-2] + (n-2)*a[n-3]}, a, {n, 0, 30}] (* _Vaclav Kotesovec_, Jul 31 2022 *)
%t nmax = 30; CoefficientList[Series[Pi*(((-BesselI[-(2/3), 2/3])*(-2 + HypergeometricPFQ[{1}, {4/3, 5/3}, 1/9]) + BesselI[1/3, 2/3]*(-2 + 2*HypergeometricPFQ[{1}, {4/3, 5/3}, 1/9] + (3/20)* HypergeometricPFQ[{2}, {7/3, 8/3}, 1/9]))/(3*Sqrt[3]))* Sqrt[1 + x]* BesselI[-(1/3), (2/3)*(1 + x)^(3/2)] + (1 + x)^2* (HypergeometricPFQ[{1}, {4/3, 5/3}, (1/9)*(1 + x)^3]/ 2) + Pi*((BesselI[2/3, 2/3]*(-2 + HypergeometricPFQ[{1}, {4/3, 5/3}, 1/9]) + BesselI[-(1/3), 2/3]*(2 - 2*HypergeometricPFQ[{1}, {4/3, 5/3}, 1/9] - (3/20)* HypergeometricPFQ[{2}, {7/3, 8/3}, 1/9]))/(3*Sqrt[3]))* Sqrt[1 + x]* BesselI[1/3, (2/3)*(1 + x)^(3/2)], {x, 0, nmax}], x] * Range[0, nmax]! // Round (* _Vaclav Kotesovec_, Jul 31 2022 *)
%o (Magma) I:=[1,1,2]; [n le 3 select I[n] else Self(n-2) + (n-3)*Self(n-3): n in [1..31]]; // _G. C. Greubel_, Jul 10 2021
%o (Sage)
%o def b(n): return 1 if (n<3) else (b(n-2) + b(n-3))/(n*(n-1))
%o [factorial(n)*b(n) for n in (0..30)] # _G. C. Greubel_, Jul 10 2021
%K nonn
%O 0,3
%A _Roger L. Bagula_, Sep 24 2006
%E Edited by _G. C. Greubel_, Jul 10 2021