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A104551
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Expansion of x/((1-x)*sqrt(1+4*x^2)).
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1
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0, 1, 1, -1, -1, 5, 5, -15, -15, 55, 55, -197, -197, 727, 727, -2705, -2705, 10165, 10165, -38455, -38455, 146301, 146301, -559131, -559131, 2145025, 2145025, -8255575, -8255575, 31861025, 31861025, -123256495, -123256495, 477823895, 477823895, -1855782325, -1855782325, 7219352975
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OFFSET
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0,6
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COMMENTS
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A transformation of the Fibonacci numbers A000045 by the Riordan array (1/sqrt(1+4*x^2), (sqrt(1+4*x^2)-1)/(2*x)).
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LINKS
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FORMULA
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G.f.: x/((1-x)*sqrt(1+4*x^2)).
a(n) = Sum_{k=0..n} (sin(Pi*k/2)+cos(Pi*k)/2+1/2)*C(k-1,(k-1)/2)*(1-(-1)^k)/2.
D-finite with recurrence: (n-1)*a(n) = (n-1)*a(n-1) - 4*(n-2)*a(n-2) + 4*(n-2)*a(n-3). - R. J. Mathar, Feb 20 2015
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MATHEMATICA
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CoefficientList[Series[x/((1-x)*Sqrt[1+4*x^2]), {x, 0, 40}], x] (* G. C. Greubel, Jan 01 2023 *)
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PROG
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(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x/((1-x)*Sqrt(1+4*x^2)) )); // G. C. Greubel, Jan 01 2023
(SageMath)
P.<x> = PowerSeriesRing(ZZ, prec)
return P( x/((1-x)*sqrt(1+4*x^2)) ).list()
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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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STATUS
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approved
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