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A165205 a(n) = C(2n-1,n) + C(2n+1,n+1) - C(0,n).

%I A165205 #15 Sep 08 2022 08:45:47
%S A165205 1,4,13,45,161,588,2178,8151,30745,116688,445094,1704794,6552378,
%T A165205 25258600,97617060,378098955,1467343305,5704370760,22210199550,
%U A165205 86595896310,338052201630,1321178419080,5168764845660,20240517205350
%N A165205 a(n) = C(2n-1,n) + C(2n+1,n+1) - C(0,n).
%C A165205 Hankel transform is A165206.
%C A165205 This sequence has the following conjectural description using cluster algebras. Consider the cyclic directed graph with n vertices as a cluster seed S of type D_n. Let S_n be the set of seeds that can be obtained from S by iterated mutation of the initial cluster variables in S. Mutations of other cluster variables are not allowed. Then the cardinality of S_n is apparently A165205(n). This can be checked from n=2 (4) to n=8 (8151). - _F. Chapoton_, Apr 03 2021
%H A165205 G. C. Greubel, <a href="/A165205/b165205.txt">Table of n, a(n) for n = 0..1000</a>
%F A165205 G.f.: (1+x)*C(x)^2/(1-x*C(x)^2) = (1+x)*(1 - sqrt(1-4*x))/(2*x* sqrt(1-4*x)), where C(x) is the g.f. of A000108.
%F A165205 Apparently (n+1)*(5*n-2) -2*(5*n+3)*(2*n-1)*a(n-1) = 0. - _R. J. Mathar_, Oct 25 2012
%t A165205 CoefficientList[Series[(1+x)*(1-Sqrt[1-4*x])/(2*x*Sqrt[1-4*x]), {x, 0, 30}], x] (* _G. C. Greubel_, Jul 18 2019 *)
%t A165205 Table[Binomial[2n-1,n]+Binomial[2n+1,n+1]-Binomial[0,n],{n,0,30}] (* _Harvey P. Dale_, Apr 28 2022 *)
%o A165205 (PARI) my(x='x+O('x^30)); Vec((1+x)*(1-sqrt(1-4*x))/(2*x*sqrt(1-4*x))) \\ _G. C. Greubel_, Jul 18 2019
%o A165205 (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1+x)*(1-Sqrt(1-4*x))/(2*x*Sqrt(1-4*x)) )); // _G. C. Greubel_, Jul 18 2019
%o A165205 (Sage) a=((1+x)*(1-sqrt(1-4*x))/(2*x*sqrt(1-4*x))).series(x, 30).coefficients(x, sparse=False); [1]+a[1:] # _G. C. Greubel_, Jul 18 2019
%K A165205 easy,nonn
%O A165205 0,2
%A A165205 _Paul Barry_, Sep 07 2009