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%I #14 Sep 08 2022 08:45:29
%S 1,-4,12,-40,124,-408,1272,-4176,13020,-42808,133096,-439344,1358872,
%T -4514800,13853040,-46469280,140945820,-479312760,1430085000,
%U -4958382960,14453014920,-51500944080,145230007440,-537922074720,1446902948184,-5662012752048,14228883685392
%N Expansion of 1/(1+4*x*c(x)), c(x) the g.f. of Catalan numbers A000108.
%C Hankel transform is (-4)^n.
%C For n>=37, all terms are negative. - _Vaclav Kotesovec_, May 30 2019
%H G. C. Greubel, <a href="/A126986/b126986.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = Sum_{k=0..n} A039599(n,k)*(-5)^k.
%F G.f.: 1/(3 - 2*sqrt(1-4*x)). - _G. C. Greubel_, May 29 2019
%F a(n) ~ -4^n / (9*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, May 30 2019
%p c:=(1-sqrt(1-4*x))/2/x: ser:=series(1/(1+4*x*c),x=0,30): seq(coeff(ser,x,n),n=0..27); # _Emeric Deutsch_, Mar 23 2007
%t CoefficientList[Series[1/(3-2*Sqrt[1-4*x]), {x,0,30}], x] (* _G. C. Greubel_, May 29 2019 *)
%o (PARI) my(x='x+O('x^30)); Vec(1/(3-2*sqrt(1-4*x))) \\ _G. C. Greubel_, May 29 2019
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/(3 - 2*Sqrt(1-4*x)) )); // _G. C. Greubel_, May 29 2019
%o (Sage) (1/(3-2*sqrt(1-4*x))).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 29 2019
%K sign
%O 0,2
%A _Philippe Deléham_, Mar 21 2007
%E More terms from _Emeric Deutsch_, Mar 23 2007
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