A178694 Numerators of coefficients of Maclaurin series for (1-x-x^2)^(-1/2).

1, 1, 7, 17, 203, 583, 3491, 10481, 254963, 779723, 4798681, 14831831, 184091359, 573076579, 3577974043, 11196388273, 561766479043, 1764905611763, 11107979665181, 35007455563451, 441899444305669, 1396202999849369

Offset: 0

Author

Clark Kimberling, Jun 04 2010

Keywords

• nonn
• frac

Comments

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

Examples

			The Maclaurin series begins with 1 + (1/2)x + (7/8)x^2 + (17/16)x^3.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A178693.

Cf. A046161 (denominators).

Programs

Magma

m:=30; R:=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

Mathematica

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 *)

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])$
makelist(a[n],n,0,50); /* Tani Akinari, Sep 14 2023 */

PARI

a(n)=numerator(I^-n*pollegendre(n,I/2)) \\ Charles R Greathouse IV, Mar 18 2017

Formula

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