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A098333
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Expansion of 1/sqrt(1 - 2x + 13x^2).
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6
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1, 1, -5, -17, 19, 211, 181, -2015, -5837, 12259, 91585, 29965, -1033955, -2347955, 7953115, 43864543, -11941037, -559875245, -942036911, 5060812717, 21502740649, -20676139991, -307241918945, -344022187613
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OFFSET
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0,3
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COMMENTS
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Central coefficients of (1 + x - 3x^2)^n.
Binomial transform of 1/sqrt(1+12x^2), or (1,0,-6,0,54,0,-540,...).
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LINKS
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FORMULA
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E.g.f.: exp(x)*BesselI(0, 2*sqrt(-3)x);
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k)*binomial(2k, k)(-3)^k;
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, k)(-3)^k.
D-finite with recurrence: n*a(n) + (-2*n+1)*a(n-1) + 13*(n-1)*a(n-2) = 0. - R. J. Mathar, Nov 24 2012
a(n) = (sqrt(13))^n*P(n,1/sqrt(13)), where P(n,x) is the Legendre polynomial of degree n. - Peter Bala, Mar 18 2018
a(n) = hypergeom([1/2 - n/2, -n/2], [1], -12). - Peter Luschny, Mar 19 2018
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MAPLE
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a := n -> hypergeom([1/2 - n/2, -n/2], [1], -12):
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MATHEMATICA
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CoefficientList[Series[1/Sqrt[1-2*x+13*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 09 2014 *)
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PROG
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(PARI) x='x+O('x^99); Vec(1/(1-2*x+13*x^2)^(1/2)) \\ Altug Alkan, Mar 18 2018
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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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