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A129369
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Expansion of 1/sqrt(1-4*x) - x/sqrt(1-4*x^2).
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5
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1, 1, 6, 18, 70, 246, 924, 3412, 12870, 48550, 184756, 705180, 2704156, 10399676, 40116600, 155114088, 601080390, 2333593350, 9075135300, 35345215180, 137846528820, 538257689684, 2104098963720, 8233430022168, 32247603683100
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OFFSET
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0,3
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COMMENTS
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LINKS
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FORMULA
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a(n) = binomial(2*n, n) - (1/2)*(1-(-1)^n)*binomial(n-1, (n-1)/2).
(n+1)^3*(n+2)*(n+3)*a(n+3) = 2*(n+1)*(n+2)^3*(2*n+1)*a(n+2) + 4*(n+1)^4*(n+3)*a(n+1) - 8*n*(n+2)^3*(2*n+1)*a(n), with a(0)=a(1) = 1, a(2) = 6. (End)
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MATHEMATICA
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CoefficientList[Series[1/Sqrt[1-4x]-x/Sqrt[1-4x^2], {x, 0, 30}], x] (* Harvey P. Dale, Feb 02 2012 *)
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PROG
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(Magma) B:=Binomial; [B(2*n, n) - (n mod 2)*B(n-1, Floor((n-1)/2)): n in [0..60]]; // G. C. Greubel, Jan 31 2024
(SageMath) [binomial(2*n, n) - (n%2)*binomial(n-1, (n-1)//2) for n in range(61)] # G. C. Greubel, Jan 31 2024
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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