A178693 Numerators of coefficients of Maclaurin series for 2 - sqrt(1 - x - x^2).

1, 1, 5, 5, 45, 95, 465, 1165, 24445, 65595, 359915, 1003315, 11342185, 32415435, 187063145, 544172445, 25508284445, 75196195795, 445774614215, 1327748661015, 15887874844835, 47715177777185, 287618252461095, 869652752181595

Offset: 0

Author

Clark Kimberling, Jun 04 2010

Keywords

• nonn
• frac

Comments

Every term after the second is a multiple of 5.

Examples

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

References

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

Links

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

Crossrefs

Cf. A178694.

Cf. A046161 (denominators).

Programs

Magma

m:=30; R:=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

Mathematica

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

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

Formula

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