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A178694
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Numerators of coefficients of Maclaurin series for (1-x-x^2)^(-1/2).
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4
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1, 1, 7, 17, 203, 583, 3491, 10481, 254963, 779723, 4798681, 14831831, 184091359, 573076579, 3577974043, 11196388273, 561766479043, 1764905611763, 11107979665181, 35007455563451, 441899444305669, 1396202999849369
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OFFSET
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0,3
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COMMENTS
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a(n) is also the numerator of I^(-n)*P_{n}(I/2) with I^2=-1 and P_{n} is the Legendre polynomial of degree n. - Alyssa Byrnes and C. Vignat, Jan 31 2013
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LINKS
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FORMULA
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G.f.: (1-x-x^2)^(-1/2) (of the series, not of this sequence).
G.f.: 1/sqrt(1-x-x^2) = G(0), where G(k)= 1 + x*(1+x)*(4*k+1)/( 4*k+2 - x*(1+x)*(4*k+2)*(4*k+3)/(x*(1+x)*(4*k+3) + 4*(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 08 2013
a(n) = numerator(b(n)), where b(n) = (1-1/n/2)*b(n-1)+(1-1/n)*b(n-2), with b(0)=1 and b(1)=1/2. - Tani Akinari, Sep 14 2023
a(n) = numerator(1/2^n*hypergeom([-n/2,(1-n)/2],[1],5)). - Gerry Martens, Sep 24 2023
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EXAMPLE
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The Maclaurin series begins with 1 + (1/2)x + (7/8)x^2 + (17/16)x^3.
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MATHEMATICA
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Numerator[CoefficientList[Series[(1-x-x^2)^(-1/2), {x, 0, 30}], x]] (* Harvey P. Dale, Oct 02 2012 *)
Table[Numerator[I^(-n)*LegendreP[n, I/2]], {n, 0, 30}] (* Alyssa Byrnes and C. Vignat, Jan 31 2013 *)
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PROG
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(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( 1/Sqrt(1-x-x^2) )); [Numerator(Factorial(n-1)*b[n]): n in [1..m]]; // G. C. Greubel, Jan 25 2019
(Maxima) b[n]:=if n<2 then 1/2^n else (1-1/n/2)*b[n-1]+(1-1/n)*b[n-2]$
a[n]:=num(b[n])$
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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