%I #37 Aug 24 2021 01:47:36
%S 1,-3,1,-5,-5,-23,-61,-203,-655,-2205,-7519,-26073,-91499,-324525,
%T -1161275,-4187605,-15202085,-55513255,-203776325,-751501075,
%U -2783025305,-10345215535,-38587318505,-144377808775,-541741418525
%N Expansion of sqrt(1-4*x)/(1+x).
%C Hankel transform is alternating sign version of A082762.
%D Joseph Edwards, Differential Calculus with Applications and Numerous Examples: An Elementary Treatise, 1886.
%F G.f.: sqrt(1-4*x)/(1+x).
%F a(n) = Sum_{k=0..n} (-1)^(n-k)*C(2k,k)/(1-2k).
%F D-finite with recurrence: n*a(n) + 3*(2-n)*a(n-1) + 2*(3-2*n)*a(n-2) = 0. - _R. J. Mathar_, Nov 16 2011.
%F G.f. A(x) = 1/G(0), where G(k) = 1 + x/(1 - (4*k+2)/((4*k+2) + (k+1)/G(k+1))); (continued fraction 3rd kind, 3-step). - _Sergei N. Gladkovskii_, Jul 24 2012
%F G.f.: 2/(1+x)/G(0), where G(k) = 1 + 1/(1 - 2*x*(2*k+1)/(2*x*(2*k+1) + (k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 24 2013
%F a(n) = binomial(2*n,n) * hypergeom([1, -n], [1/2], 5/4). - _Vladimir Reshetnikov_, Oct 02 2016
%F a(n) ~ -2^(2*n+1) / (5*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 03 2016
%t CoefficientList[Series[Sqrt[1-4x]/(1+x),{x,0,30}],x] (* _Harvey P. Dale_, Jul 20 2011 *)
%t Table[FullSimplify[(-1)^n*Sqrt[5] - 1/(1+2*n)*Binomial[2*(1+n),1+n] * Hypergeometric2F1[1,1/2+n,2+n,-4]],{n,0,20}] (* _Vaclav Kotesovec_, Jan 31 2014 *)
%t Table[Binomial[2 n, n] Hypergeometric2F1[1, -n, 1/2, 5/4], {n, 0, 30}] (* _Vladimir Reshetnikov_, Oct 02 2016 *)
%o (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*binomial(2*k,k)/(1-2*k)); \\ _Michel Marcus_, Oct 03 2016
%Y Cf. A082762.
%K easy,sign
%O 0,2
%A _Paul Barry_, Nov 03 2010