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%I #19 Sep 08 2022 08:45:29
%S 1,-2,2,-4,2,-12,-12,-72,-190,-700,-2308,-8120,-28364,-100856,-360792,
%T -1301904,-4727358,-17268636,-63405012,-233885784,-866327748,
%U -3220976616,-12016209192,-44966763504,-168750724428,-634935132312,-2394717424552,-9051945482032
%N Expansion of 1/(1+2*x*c(x)), c(x) the g.f. of Catalan numbers A000108.
%C Hankel transform is (-2)^n.
%C Hankel transform omitting first term is (-2)^n omitting first term. Hankel transform omitting first two terms is 2*(-1)^n*A014480(n). - _Michael Somos_, May 16 2022
%H G. C. Greubel, <a href="/A126984/b126984.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = Sum_{k=0..n} A039599(n,k)*(-3)^k.
%F G.f.: 1/(2 - sqrt(1-4*x)). - _G. C. Greubel_, May 28 2019
%F (-1)^n*a(n) = A268600(n) - A268601(n). - _Michael Somos_, May 16 2022
%p c:=(1-sqrt(1-4*x))/2/x: ser:=series(1/(1+2*x*c),x=0,32): seq(coeff(ser,x,n),n=0..30); # _Emeric Deutsch_, Mar 24 2007
%t CoefficientList[Series[1/(2-Sqrt[1-4*x]), {x,0,30}], x] (* _G. C. Greubel_, May 28 2019 *)
%o (PARI) my(x='x+O('x^30)); Vec(1/(2-sqrt(1-4*x))) \\ _G. C. Greubel_, May 28 2019
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/(2-Sqrt(1-4*x)) )); // _G. C. Greubel_, May 28 2019
%o (Sage) (1/(2-sqrt(1-4*x))).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 28 2019
%Y Cf. A000108, A014480, A039599, A268600, A268601.
%K sign
%O 0,2
%A _Philippe Deléham_, Mar 21 2007
%E Corrected and extended by _Emeric Deutsch_, Mar 24 2007
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