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A370491
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The numerators of a series that converges to the Omega constant (A030178) obtained using Whittaker's root series formula.
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1
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1, 1, -1, -5, 19, -3, -10187, 146847, 3268961, -211632497, 393324007, 5402916117, -3884618921299, -774402304798329, 148294948981707557, -3311395903665985169, -43463254022673425965, 14469962812566878696039, 6554498075974546253080309, -3074689522272735111427973673
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OFFSET
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1,4
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COMMENTS
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Whittaker's root series formula is applied to 1 - 2x + x^2/2! - x^3/3! + x^4/4! - x^5/5! + x^6/6! - ..., which is the Taylor expansion of -x + e^(-x). We obtain the following infinite series that converges to the Omega constant (LambertW(1)): LambertW(1) = 1/2 + 1/14 - 1/259 - 5/9657 + 19/200187 - 3/18671081 ... . The sequence is formed by the numerators of the infinite series.
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LINKS
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FORMULA
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For n > 1, a(n) is the numerator of the simplified fraction -det ToeplitzMatrix((c(2),c(1),c(0),0,0,...,0),(c(2),c(3),c(4),...,c(n)))/(det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n-1)))*det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n)))), where c(0)=1, c(1)=-2, c(n) = (-1)^n/n!.
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EXAMPLE
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a(1) is the numerator of -1/-2 = 1/2.
a(2) is the numerator of -(1/2)/((-2)*det ToeplitzMatrix((-2,1),(-2,1/2!)) = -(1/2)/((-2)(7/2)) = 1/14.
a(3) is the numerator of -det ToeplitzMatrix((1/2!,-2),(1/2!,-1/3!))/(det ToeplitzMatrix((-2,1),(-2,1/2!)*det ToeplitzMatrix((-2,1,0),(-2,1/2!,-1/3!))) = -(-1/12)/((7/2)(-37/6)) = -1/259.
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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