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%I #36 Sep 24 2023 10:50:14
%S 1,1,7,17,203,583,3491,10481,254963,779723,4798681,14831831,184091359,
%T 573076579,3577974043,11196388273,561766479043,1764905611763,
%U 11107979665181,35007455563451,441899444305669,1396202999849369
%N Numerators of coefficients of Maclaurin series for (1-x-x^2)^(-1/2).
%C 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
%H G. C. Greubel, <a href="/A178694/b178694.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: (1-x-x^2)^(-1/2) (of the series, not of this sequence).
%F 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
%F 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
%F a(n) = numerator(1/2^n*hypergeom([-n/2,(1-n)/2],[1],5)). - _Gerry Martens_, Sep 24 2023
%e The Maclaurin series begins with 1 + (1/2)x + (7/8)x^2 + (17/16)x^3.
%t Numerator[CoefficientList[Series[(1-x-x^2)^(-1/2),{x,0,30}],x]] (* _Harvey P. Dale_, Oct 02 2012 *)
%t Table[Numerator[I^(-n)*LegendreP[n, I/2]], {n, 0, 30}] (* _Alyssa Byrnes_ and C. Vignat, Jan 31 2013 *)
%o (PARI) a(n)=numerator(I^-n*pollegendre(n,I/2)) \\ _Charles R Greathouse IV_, Mar 18 2017
%o (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
%o (Maxima) b[n]:=if n<2 then 1/2^n else (1-1/n/2)*b[n-1]+(1-1/n)*b[n-2]$
%o a[n]:=num(b[n])$
%o makelist(a[n],n,0,50); /* _Tani Akinari_, Sep 14 2023 */
%Y Cf. A178693.
%Y Cf. A046161 (denominators).
%K nonn,frac
%O 0,3
%A _Clark Kimberling_, Jun 04 2010
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