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A277432
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E.g.f.: sinh(sqrt(2)*x)/(sqrt(2)*(1-x)).
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2
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0, 1, 2, 8, 32, 164, 984, 6896, 55168, 496528, 4965280, 54618112, 655417344, 8520425536, 119285957504, 1789289362688, 28628629803008, 486686706651392, 8760360719725056, 166446853674776576, 3328937073495531520, 69907678543406162944, 1537968927954935584768
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = (Gamma(n+1, sqrt(2))*exp(sqrt(2)) - Gamma(n+1, -sqrt(2))/exp(sqrt(2))) / (2*sqrt(2)).
a(n) ~ sqrt(Pi)*sinh(sqrt(2))*n^(n+1/2)*exp(-n).
D-finite with recurrence: a(n) = n*a(n-1) + 2*a(n-2) - 2*(n-2)*a(n-3).
Gamma(n+1, sqrt(2))*exp(sqrt(2)) = A277431(n) + sqrt(2)*a(n).
Gamma(n+1, -sqrt(2))/exp(sqrt(2)) = A277431(n) - sqrt(2)*a(n).
For n > 0, a(2*n) = 2*n*a(2*n-1).
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MAPLE
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f:= gfun:-rectoproc({a(n) = n*a(n-1) + 2*a(n-2) - 2*(n-2)*a(n-3), a(0)=0, a(1)=1, a(2)=2}, a(n), remember):
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MATHEMATICA
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Round@Table[(Gamma[n + 1, Sqrt[2]] Exp[Sqrt[2]] - Gamma[n + 1, -Sqrt[2]]/Exp[Sqrt[2]])/(2 Sqrt[2]), {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster *)
Expand@Table[SeriesCoefficient[Sinh[Sqrt[2] x]/(Sqrt[2] (1 - x)), {x, 0, n}] n!, {n, 0, 20}]
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PROG
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(PARI) x='x+O('x^30); concat([0], round(Vec(serlaplace(sinh(sqrt(2)*x)/( sqrt(2)*(1-x)))))) \\ G. C. Greubel, Aug 19 2018
(Magma) I:=[1, 2, 8]; [0] cat [n le 3 select I[n] else n*Self(n-1) + 2*Self(n-2) - 2*(n-2)*Self(n-3): n in [1..30]]; // G. C. Greubel, Aug 19 2018
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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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