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A126984 Expansion of 1/(1+2*x*c(x)), c(x) the g.f. of Catalan numbers A000108.

%I A126984 #28 Jun 24 2025 13:22:28
%S A126984 1,-2,2,-4,2,-12,-12,-72,-190,-700,-2308,-8120,-28364,-100856,-360792,
%T A126984 -1301904,-4727358,-17268636,-63405012,-233885784,-866327748,
%U A126984 -3220976616,-12016209192,-44966763504,-168750724428,-634935132312,-2394717424552,-9051945482032
%N A126984 Expansion of 1/(1+2*x*c(x)), c(x) the g.f. of Catalan numbers A000108.
%C A126984 Hankel transform is (-2)^n.
%C A126984 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 A126984 G. C. Greubel, <a href="/A126984/b126984.txt">Table of n, a(n) for n = 0..1000</a>
%F A126984 a(n) = Sum_{k=0..n} A039599(n,k)*(-3)^k.
%F A126984 G.f.: 1/(2 - sqrt(1-4*x)). - _G. C. Greubel_, May 28 2019
%F A126984 (-1)^n*a(n) = A268600(n) - A268601(n). - _Michael Somos_, May 16 2022
%F A126984 D-finite with recurrence 3*n*a(n) +2*(-4*n+9)*a(n-1) +8*(-2*n+3)*a(n-2)=0. - _R. J. Mathar_, Nov 22 2024
%F A126984 a(n) = Sum_{k = 0..n} A009766(n-1, k)*(-2)^(n-k) for n >= 1. - _Peter Bala_, Jun 18 2025
%p A126984 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 A126984 CoefficientList[Series[1/(2-Sqrt[1-4*x]), {x,0,30}], x] (* _G. C. Greubel_, May 28 2019 *)
%o A126984 (PARI) my(x='x+O('x^30)); Vec(1/(2-sqrt(1-4*x))) \\ _G. C. Greubel_, May 28 2019
%o A126984 (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/(2-Sqrt(1-4*x)) )); // _G. C. Greubel_, May 28 2019
%o A126984 (Sage) (1/(2-sqrt(1-4*x))).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 28 2019
%Y A126984 Cf. A000108, A014480, A039599, A268600, A268601.
%K A126984 sign,easy
%O A126984 0,2
%A A126984 _Philippe Deléham_, Mar 21 2007
%E A126984 Corrected and extended by _Emeric Deutsch_, Mar 24 2007