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A098338
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Expansion of 1/sqrt(1-6x+13x^2).
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1
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1, 3, 7, 9, -21, -207, -911, -2769, -5213, 2457, 74997, 400491, 1409109, 3323583, 2219343, -27453951, -186624333, -750905127, -2088947819, -2955863589, 8506703569, 86421384387, 401183114163, 1280139325101, 2522745571021
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OFFSET
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0,2
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COMMENTS
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Central coefficients of (1+3x-x^2)^n.
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LINKS
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FORMULA
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E.g.f.: exp(3*x)*BesselI(0, 2*I*x), I=sqrt(-1).
a(n) = Sum{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, k)*3^n*(-9)^(-k).
a(n) = Sum{k=0..floor(n/2)} binomial(n, 2k)*binomial(2k, k)*3^n*(-9)^(-k).
D-finite with recurrence: n*a(n) +3*(1-2*n)*a(n-1) +13*(n-1)*a(n-2)=0. - R. J. Mathar, Sep 26 2012
Recurrence follows from the differential equation (13x-3) g(x) + (13x^2-6x+1) g'(x) = 0 satisfied by the generating function. - Robert Israel, Mar 02 2017
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MAPLE
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f:= gfun:-rectoproc({(13*n+13)*a(n)+(-9-6*n)*a(n+1)+(n+2)*a(n+2), a(0)=1, a(1)=3}, a(n), remember):
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MATHEMATICA
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CoefficientList[Series[1/Sqrt[1-6*x+13*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 29 2013 *)
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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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