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A178694 Numerators of coefficients of Maclaurin series for (1-x-x^2)^(-1/2). 4
1, 1, 7, 17, 203, 583, 3491, 10481, 254963, 779723, 4798681, 14831831, 184091359, 573076579, 3577974043, 11196388273, 561766479043, 1764905611763, 11107979665181, 35007455563451, 441899444305669, 1396202999849369 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

LINKS

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

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

EXAMPLE

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

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 *)

PROG

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

(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

CROSSREFS

Cf. A178693.

Cf. A046161 (denominators).

Sequence in context: A214149 A147643 A061159 * A140122 A092240 A110120

Adjacent sequences:  A178691 A178692 A178693 * A178695 A178696 A178697

KEYWORD

nonn,frac

AUTHOR

Clark Kimberling, Jun 04 2010

STATUS

approved

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Last modified December 3 18:15 EST 2020. Contains 338908 sequences. (Running on oeis4.)