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A123024
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a(n) = n!*b(n) where b(n) = (b(n-2) + b(n-3))/(n*(n-1)), b(0) = b(1) = b(2) = 1.
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1
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1, 1, 2, 2, 4, 8, 12, 28, 60, 112, 284, 652, 1404, 3776, 9228, 22028, 62092, 160448, 414540, 1216012, 3302604, 9092272, 27622844, 78446956, 227652828, 713772368, 2110379772, 6405093068, 20668461340, 63385346912, 200011067244
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OFFSET
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0,3
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REFERENCES
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Richard Bronson, Schaum's Outline of Modern Introductory Differential Equations, MacGraw-Hill, New York,1973, page 107, solved problem 19.15.
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LINKS
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FORMULA
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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-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]
(End)
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.
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)
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MATHEMATICA
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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}]
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 *)
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 *)
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PROG
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(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
(Sage)
def b(n): return 1 if (n<3) else (b(n-2) + b(n-3))/(n*(n-1))
[factorial(n)*b(n) for n in (0..30)] # G. C. Greubel, Jul 10 2021
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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