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A104629 Expansion of (1-2*x-sqrt(1-4*x))/(x^2 * (1+2*x+sqrt(1-4*x))). 5

%I

%S 1,2,6,18,57,186,622,2120,7338,25724,91144,325878,1174281,4260282,

%T 15548694,57048048,210295326,778483932,2892818244,10786724388,

%U 40347919626,151355847012,569274150156,2146336125648,8110508473252

%N Expansion of (1-2*x-sqrt(1-4*x))/(x^2 * (1+2*x+sqrt(1-4*x))).

%C Diagonal sums of A039598. a(n)=A000957(n+3).

%H G. C. Greubel, <a href="/A104629/b104629.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = (1 + Sum_{k=0..n+2} C(k)*(-2)^k)/(8*(-2)^n), where C(n) = Catalan numbers.

%F Conjecture: 2*(n+3)*a(n) +(-7*n-9)*a(n-1) +2*(-2*n-3)*a(n-2)=0. - _R. J. Mathar_, Oct 30 2014

%t CoefficientList[Series[((1-2x-Sqrt[1-4x])/(1+2x+Sqrt[1-4x]))/x^2,{x,0,30}],x] (* _Harvey P. Dale_, Jul 23 2016 *)

%t Table[(1 + Sum[CatalanNumber[n]*(-2)^k, {k,0,n+2}])/(8*(-2)^n), {n,0,30}] (* _G. C. Greubel_, Aug 12 2018 *)

%o (PARI) x='x+O('x^30); Vec((1-2*x-sqrt(1-4*x))/(x^2*(1+2*x+sqrt(1-4*x)))) \\ _G. C. Greubel_, Aug 12 2018

%o (PARI) for(n=0,30, print1((1 + sum(k=0,n+2, (-2)^k*binomial(2*k, k)/(k+1)))/(8*(-2)^n), ", ")) \\ _G. C. Greubel_, Aug 12 2018

%o (MAGMA) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-2*x-Sqrt(1-4*x))/(x^2*(1+2*x+Sqrt(1-4*x))))); // _G. C. Greubel_, Aug 12 2018

%Y Cf. A064310.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Mar 17 2005

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Last modified February 20 20:53 EST 2019. Contains 320345 sequences. (Running on oeis4.)