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A098335
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Expansion of 1/sqrt(1-4x+8x^2).
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7
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1, 2, 2, -4, -26, -68, -76, 184, 1222, 3308, 3772, -9656, -64676, -177448, -203992, 536176, 3607622, 9968972, 11510636, -30723416, -207302156, -575382392, -666187432, 1796105744, 12142184476, 33803271032
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OFFSET
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0,2
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COMMENTS
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Central coefficients of (1+2x-x^2)^n. Binomial transform of A098331.
Diagonal of rational function 1/(1 - (x^2 + 2*x*y - y^2)). - Gheorghe Coserea, Aug 04 2018
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LINKS
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FORMULA
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E.g.f. : exp(2*x)*BesselI(0, 2*I*x), I=sqrt(-1);
a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*binomial(n-k,k)*2^n*(-4)^(-k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*binomial(2(n-k),k)*(-2)^k. (End)
It appears that a(j+2) = (2*(2*j+1)*a(j+1))/(j+1)-(8*j*a(j))/(j+1), in case of re-indexing from 0 to 1. - Alexander R. Povolotsky, Aug 22 2012
D-finite with recurrence: a(n+2) = ((4*n+6)*a(n+1) - 8*(n+1)*a(n))/(n+2); a(0)=1,a(1)=2. - Sergei N. Gladkovskii, Aug 22 2012
a(n) = 2^n*_2F_1(1/2-n/2, -n/2, 1, -1), where _2F_1(a,b;c;x) is the hypergeometric function. - Alexander R. Povolotsky, Aug 22 2012
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MATHEMATICA
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CoefficientList[Series[1/Sqrt[1 - 4*x + 8*x^2], {x, 0, 50}], x] (* G. C. Greubel, Feb 19 2017 *)
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PROG
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(PARI) x='x+O('x^25); Vec(1/sqrt(1 - 4*x + 8*x^2)) \\ G. C. Greubel, Feb 19 2017
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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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