A178693 - OEIS

%I #16 Sep 08 2022 08:45:54

%S 1,1,5,5,45,95,465,1165,24445,65595,359915,1003315,11342185,32415435,

%T 187063145,544172445,25508284445,75196195795,445774614215,

%U 1327748661015,15887874844835,47715177777185,287618252461095,869652752181595

%N Numerators of coefficients of Maclaurin series for 2 - sqrt(1 - x - x^2).

%C Every term after the second is a multiple of 5.

%D M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 41.

%H G. C. Greubel, <a href="/A178693/b178693.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: 2 - sqrt(1 - x - x^2) for the fractions (not the numerators).

%e The Maclaurin series begins with 1 + (1/2)*x + (5/8)*x^2 + (5/16)*x^3 + ....

%t Numerator[CoefficientList[Series[2-Sqrt[1-x-x^2], {x, 0, 30}], x]] (* _G. C. Greubel_, Jan 25 2019 *)

%o (PARI) my(x='x+O('x^30)); v=Vec( 2-sqrt(1-x-x^2) ); vector(#v, n, numerator(v[n])) \\ _G. C. Greubel_, Jan 25 2019

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( 2-Sqrt(1-x-x^2) )); [Numerator(Factorial(n-1)*b[n]): n in [1..m]]; // _G. C. Greubel_, Jan 25 2019

%Y Cf. A178694.

%Y Cf. A046161 (denominators).

%K nonn,frac

%O 0,3

%A _Clark Kimberling_, Jun 04 2010