A126121 Numerators of sequence of fractions with e.g.f. sqrt(1+x)/(1-x)^2.

1, 5, 31, 255, 2577, 31245, 439695, 7072695, 127699425, 2562270165, 56484554175, 1358576240175, 35374065613425, 992016072172125, 29792674421484975, 954480422711190375, 32479589325536978625, 1170329273010701929125, 44502357662442514209375, 1781390379962467540641375

Offset: 0

Author

N. J. A. Sloane, Mar 22 2007

Keywords

• nonn
• frac

Comments

Denominators are successive powers of 2.

Examples

			The fractions are 1, 5/2, 31/4, 255/8, 2577/16, 31245/32, 439695/64, ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A003148, A001174, A126115, A126119.

Programs

Mathematica

With[{nn=20},Numerator[CoefficientList[Series[Sqrt[1+x]/(1-x)^2,{x, 0, nn}], x] Range[0,nn]!]] (* Harvey P. Dale, Jan 29 2016 *)

PARI

x='x+O('x^25); Vec(serlaplace(sqrt(1+2*x)/(1-2*x)^2)) \\ G. C. Greubel, May 25 2017

Formula

E.g.f.: 1/G(0) where G(k) = 1 - 4*x/(1 + x/(1 - x - (2*k+1)/( 2*k+1 - 4*(k+1)*x/G(k+1)))); (continued fraction, 3rd kind, 4-step). - Sergei N. Gladkovskii, Jul 28 2012

From Benedict W. J. Irwin, May 19 2016: (Start)

E.g.f.: sqrt(1+2*x)/(1-2*x)^2.

a(n) = (-1)^(n+1)*2^(n-1)*(n-3/2)!*2F1(2,-n;(3/2)-n;-1)/sqrt(Pi).

(End)

D-finite with recurrence a(n) -5*a(n-1) -2*(2*n-1)*(n-1)*a(n-2)=0. - R. J. Mathar, Feb 08 2021