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A127363
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a(n) = Sum_{k=0..n} C(n,floor(k/2))*(-4)^(n-k).
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4
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1, -3, 14, -57, 246, -1038, 4424, -18777, 79846, -339258, 1442004, -6128202, 26045436, -110691948, 470442924, -1999378137, 8497365126, -36113785698, 153483619604, -652305322542, 2772297736276, -11782265148228, 50074627320864, -212817165231882, 904472953925596
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OFFSET
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0,2
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COMMENTS
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Hankel transform is 5^n. In general, for r>=0, the sequence given by Sum_{k=0..n} C(n,floor(k/2))*(-r)^(n-k) has Hankel transform (r+1)^n. The sequence is the image of the sequence with g.f. (1+x)/(1+4x) under the Chebyshev mapping g(x)->(1/sqrt(1-4x^2))g(xc(x^2)), where c(x) is the g.f. of the Catalan numbers A000108.
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LINKS
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FORMULA
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G.f.: (1/sqrt(1-4x^2))(1+x*c(x^2))/(1+4*x*c(x^2)).
Conjecture: 4*n*a(n) +(17*n-8)*a(n-1) +2*(-8*n-1)*a(n-2) +68*(-n+2)*a(n-3)=0. - R. J. Mathar, Nov 24 2012
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MATHEMATICA
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CoefficientList[Series[1/Sqrt[1-4*x^2] * (1+x*(1-Sqrt[1-4*x^2]) / (2*x^2)) / (1+4*x*(1-Sqrt[1-4*x^2])/(2*x^2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)
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CROSSREFS
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KEYWORD
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easy,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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