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A158499
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Expansion of (1+sqrt(1-4x))/(2-4x).
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1
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1, 1, 1, 0, -5, -24, -90, -312, -1053, -3536, -11934, -40664, -140114, -488240, -1719380, -6113200, -21921245, -79200160, -288045110, -1053728920, -3874721030, -14313562480, -53093391980, -197669347600, -738398308850, -2766700765024
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OFFSET
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0,5
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COMMENTS
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Hankel transform is A056594 with g.f. 1/(1+x^2).
Row sums of the Riordan array (sqrt(1-4x)/(1-2x),xc(x)^2), c(x) the g.f. of A000108.
The inverse Catalan transform yields A146559. - R. J. Mathar, Mar 20 2009
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LINKS
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Matthew House, Table of n, a(n) for n = 0..1669
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
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FORMULA
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a(n) = Sum_{k=0..n} binomial(2k,k)*A158495(n-k).
Conjecture: n*a(n) +6*(1-n)*a(n-1) +4*(2*n-3)*a(n-2)=0. - R. J. Mathar, Nov 14 2011
This conjecture has been proven. - Matthew House, Nov 08 2015
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MATHEMATICA
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CoefficientList[ Series[(1 + Sqrt[1 - 4x])/(2 - 4x), {x, 0, 26}], x] (* Robert G. Wilson v, Nov 08 2015 *)
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PROG
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(PARI) x='x+O('x^33); Vec(((1-4*x)+sqrt(1-4*x))/(2*(1-2*x)*sqrt(1-4*x))) \\ Altug Alkan, Nov 08 2015
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CROSSREFS
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Sequence in context: A089095 A220316 A220339 * A074085 A145914 A066316
Adjacent sequences: A158496 A158497 A158498 * A158500 A158501 A158502
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KEYWORD
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easy,sign
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AUTHOR
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Paul Barry, Mar 20 2009
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EXTENSIONS
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Name edited by Matthew House, Nov 08 2015
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STATUS
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approved
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