A126121 - OEIS

%I #26 Feb 08 2021 07:15:22

%S 1,5,31,255,2577,31245,439695,7072695,127699425,2562270165,

%T 56484554175,1358576240175,35374065613425,992016072172125,

%U 29792674421484975,954480422711190375,32479589325536978625,1170329273010701929125,44502357662442514209375,1781390379962467540641375

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

%C Denominators are successive powers of 2.

%H G. C. Greubel, <a href="/A126121/b126121.txt">Table of n, a(n) for n = 0..400</a>

%F 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

%F From _Benedict W. J. Irwin_, May 19 2016: (Start)

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

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

%F (End)

%F 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

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

%t 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 *)

%o (PARI) x='x+O('x^25); Vec(serlaplace(sqrt(1+2*x)/(1-2*x)^2)) \\ _G. C. Greubel_, May 25 2017

%Y Cf. A003148, A001174, A126115, A126119.

%K nonn,frac

%O 0,2

%A _N. J. A. Sloane_, Mar 22 2007