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A071357
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Expansion of (1 - 4*x - (1-2*x)*sqrt(1-4*x-4*x^2))/(8*x^3).
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2
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0, 1, 4, 16, 64, 260, 1072, 4480, 18944, 80928, 348800, 1515008, 6625280, 29147456, 128918272, 572928000, 2557100032, 11457170944, 51514963968, 232370167808, 1051235287040, 4768568354816, 21684663148544, 98835356778496, 451433970008064
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = (1/Pi)*Integral_{x=2-2*sqrt(2)..2+2*sqrt(2)} x^n*sqrt(-x^2+4x+4)*(x-2)/8. - Paul Barry, Sep 16 2006
a(n) = Sum_{k=0..n} 2^(n-k)*binomial(n,k)*2^((k-1)/2)*C((k-1)/2+1)*(1-(-1)^k)/2, where C(n)=A000108(n). - Paul Barry, Sep 16 2006
D-finite with recurrence: a(n) = (1/(n+3))*((6*n+8)*a(n-1) - (4*n-4)*a(n-2) - (8*n-16)*a(n-3)) for n > 2, with a(0)=0, a(1)=1, a(2)=4. - Tani Akinari, Jul 04 2013
a(n) ~ 2^(n + 1/4) * (1 + sqrt(2))^(n + 3/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 03 2019
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MATHEMATICA
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CoefficientList[Series[(1-4x-(1-2x)Sqrt[1-4x-4x^2])/(8x^3), {x, 0, 30}], x] (* Harvey P. Dale, Aug 09 2016 *)
Table[Sum[2^(n-k) * Binomial[n, k] * 2^((k-1)/2) * CatalanNumber[(k+1)/2] * (1 - (-1)^k)/2, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 03 2019 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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