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%I #45 Mar 08 2024 08:11:08
%S 1,0,-2,-8,-26,-80,-244,-752,-2362,-7584,-24892,-83376,-284324,
%T -984672,-3455144,-12259168,-43908026,-158531392,-576352364,
%U -2107982128,-7750490636,-28629222112,-106190978264,-395347083808,-1476813394916,-5533435084480,-20790762971864,-78316232088032
%N Expansion of sqrt(1 - 4*x)/(1 - 2*x).
%C Hankel transform is 2^n*(-1)^binomial(n+1, 2) = A120617(n). - _Paul Barry_, Feb 08 2008
%H G. C. Greubel, <a href="/A126966/b126966.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = -Sum_{j=0..n} ( 2^j*binomial(2n-2j, n-j)/(2n-2j-1) ). - _Emeric Deutsch_, Mar 25 2007
%F D-finite with recurrence: n*a(n) + 6*(1-n)*a(n-1) + 4*(2*n-3)*a(n-2) = 0. - _R. J. Mathar_, Nov 14 2011, corrected Feb 17 2020
%F a(n) ~ -4^n/(sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jun 29 2013
%F a(n) = 2^n*i + CatalanNumber(n)*hypergeom([1, n + 1/2], [n + 2], 2). - _Peter Luschny_, Aug 04 2020
%F a(n) = A028329(n) - A082590(n). - _Mélika Tebni_, Mar 08 2024
%p a := n -> -add(2^j*binomial(2*n-2*j,n-j)/(2*n-2*j-1), j=0..n):
%p seq(a(n),n=0..30); # _Emeric Deutsch_, Mar 25 2007
%p # second Maple program:
%p CatalanNumber := n -> binomial(2*n, n)/(n+1):
%p a := n -> 2^n*I + CatalanNumber(n)*simplify(hypergeom([1, n + 1/2], [n + 2], 2)):
%p seq(a(n), n=0..26); # _Peter Luschny_, Aug 04 2020
%p # third program:
%p A126966 := n -> 2*binomial(2*n, n) - add(2^(n-k)*binomial(2*k,k), k=0..n):
%p seq(A126966(n), n = 0 .. 27); # _Mélika Tebni_, Mar 08 2024
%t CoefficientList[Series[Sqrt[1-4*x]/(1-2*x), {x,0,30}], x] (* _G. C. Greubel_, Jan 31 2017 *)
%o (PARI) Vec(sqrt(1-4*x)/(1-2*x) + O(x^30)) \\ _G. C. Greubel_, Jan 31 2017
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( Sqrt(1-4*x)/(1-2*x) )); // _G. C. Greubel_, Jan 29 2020
%o (Sage)
%o def A126966_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( sqrt(1-4*x)/(1-2*x) ).list()
%o A126966_list(30) # _G. C. Greubel_, Jan 29 2020
%o (GAP) List([0..30], n-> (-1)*Sum([0..n], j-> 2^j*Binomial(2*(n-j), n-j)/(2*(n-j) -1) )); # _G. C. Greubel_, Jan 29 2020
%Y Cf. A000108, A028329, A082590, A126967.
%K sign
%O 0,3
%A _N. J. A. Sloane_, Mar 22 2007
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