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A206300
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Expand the real root of y^3 - y + x in powers of x, then multiply coefficient of x^n by -4^n to get integers.
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3
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-1, 2, 6, 32, 210, 1536, 12012, 98304, 831402, 7208960, 63740820, 572522496, 5209363380, 47915728896, 444799488600, 4161823309824, 39209074920090, 371626340253696, 3541117629057540
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OFFSET
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0,2
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COMMENTS
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Also coefficients of the series S(u) for which (-sqrt(3u))*S converges to the larger of the two real roots of x^3 - 3ux + 4u for u >= 4. Specifically, S(u)=Sum_{n>=0} a(n)/(27*u)^(n/2). - Dixon J. Jones, Jun 24 2021
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REFERENCES
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G. E. Andrews, Number Theory, 1971, Dover Publications New York, pp. 41 - 43.
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LINKS
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FORMULA
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G.f.: - (1/x) * Revert( x*sqrt(c(4*x)) ), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108 and sqrt(c(4*x)) is the g.f. of A048990. - Peter Bala, Mar 05 2020
a(n) = 2^(2*n - 1) Gamma((3*n - 1)/2)/(Gamma((n + 1)/2)*n!).
a(n) = (2^(2*n - 1)*((n + 1)/2)_(n-1))/n!, where (x)_k is the Pochhammer symbol. (End)
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MATHEMATICA
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p[x_] = y /. Solve[y^3 - y + x == 0, y][[1]]
b = Table[-4^n*FullSimplify[ExpandAll[SeriesCoefficient[ Series[p[x], {x, 0, 30}], n]]], {n, 0, 30}]
Clear[a]; a=Table[2^(2n - 1) Gamma[(3n - 1)/2]/(Gamma[(n + 1)/2]n!), {n, 0, 20}]
Clear[a]; a=Table[2^(2n - 1) Pochhammer[(n + 1)/2, (n-1)]/n!, {n, 0, 20}] (* End *)
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PROG
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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