This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A137720 #23 Sep 26 2024 22:20:06
%S A137720 1,1,1,-1,-13,-67,-285,-1119,-4215,-15505,-56239,-202309,-724499,
%T A137720 -2589521,-9254363,-33111969,-118725597,-426892131,-1539965973,
%U A137720 -5575175319,-20260052337,-73908397851,-270657727593,-994938310059
%N A137720 Expansion of sqrt(1-4*x)/(1-3*x).
%C A137720 Hankel transform is A120617. In general, sqrt(1-4*x)/(1-k*x) has Hankel transform with g.f. of (1-2*x)/(1+2*(k+2)*x+4*x^2).
%H A137720 G. C. Greubel, <a href="/A137720/b137720.txt">Table of n, a(n) for n = 0..1000</a>
%F A137720 a(n) = Sum_{k=0..n} 3^k*C(2*n-2*k,n-k)/(1-(2*n-2*k)).
%F A137720 D-finite with recurrence: n*a(n) + (6-7*n)*a(n-1) + 6*(2*n-3)*a(n-2) = 0. - _R. J. Mathar_, Nov 16 2011
%F A137720 a(n) ~ -2^(2*n+1) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Jul 31 2014
%F A137720 a(n) = (-1)^n * A157674(2*n+1). - _Vaclav Kotesovec_, Jul 31 2014
%t A137720 CoefficientList[Series[Sqrt[1-4*x]/(1-3*x), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Jul 31 2014 *)
%t A137720 FullSimplify[Table[I*3^(-1/2+n) + 2^(1+2*n)*Gamma[1/2+n] * Hypergeometric2F1Regularized[1, 1/2+n, 2+n, 4/3]/(3*Sqrt[Pi]), {n, 0, 20}]] (* _Vaclav Kotesovec_, Jul 31 2014 *)
%o A137720 (PARI) x='x+O('x^50); Vec(sqrt(1-4*x)/(1-3*x)) \\ _G. C. Greubel_, Mar 21 2017
%Y A137720 Cf. A126966, A106191, A157674.
%K A137720 easy,sign
%O A137720 0,5
%A A137720 _Paul Barry_, Feb 08 2008