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%I #26 Apr 27 2023 06:11:20
%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.
%H G. C. Greubel, <a href="/A104629/b104629.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = A000957(n+3).
%F a(n) = (1 + Sum_{k=0..n+2} C(k)*(-2)^k)/(8*(-2)^n), where C(n) = Catalan numbers.
%F D-finite with recurrence: 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 [Verified by _Georg Fischer_, Apr 27 2023]
%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
%o (Python)
%o from itertools import count, islice
%o def A104629_gen(): # generator of terms
%o a, c = 0, 1
%o for n in count(1):
%o yield (a:=(c:=c*((n<<2)+2)//(n+2))-a>>1)
%o A104629_list = list(islice(A104629_gen(),20)) # _Chai Wah Wu_, Apr 26 2023
%Y Cf. A000108, A000957, A039598, A064310.
%Y Partial sums of A122920.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Mar 17 2005
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