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A026773 a(n) = T(2n-1,n-1), T given by A026769. Also T(2n+1,n+1), T given by A026780.

%I A026773 #27 Aug 06 2024 00:02:55
%S A026773 1,4,17,76,352,1674,8129,40156,201236,1020922,5234660,27089726,
%T A026773 141335846,742712598,3927908193,20891799036,111688381228,599841215226,
%U A026773 3234957053984,17512055200470,95125188934942,518340392855286,2832580291316092,15520177744727766
%N A026773 a(n) = T(2n-1,n-1), T given by A026769. Also T(2n+1,n+1), T given by A026780.
%H A026773 Alois P. Heinz, <a href="/A026773/b026773.txt">Table of n, a(n) for n = 1..1000</a>
%F A026773 From _Vladeta Jovovic_, Nov 23 2003: (Start)
%F A026773 a(n) = A006318(n) - A000108(n).
%F A026773 G.f.: (sqrt(1-4*x) - sqrt(1-6*x+x^2))/(2*x) -1/2. (End)
%F A026773 From _Paul Barry_, May 19 2005: (Start)
%F A026773 a(n) = Sum_{k=0..n} C(n+k+1, n+1)*C(n+1, k)/(k+1).
%F A026773 a(n) = Sum_{k=0..n+1} C(n+2, k)*C(n+k, n+1)/(n+2). (End)
%F A026773 D-finite with recurrence n*(n+1)*a(n) -n*(11*n-7)*a(n-1) +(37*n^2-95*n+54)*a(n-2) +(-49*n^2+269*n-354)*a(n-3) +6*(9*n^2-71*n+138)*a(n-4) -4*(2*n-9)*(n-5)*a(n-5)=0. - _R. J. Mathar_, Aug 05 2021
%p A026773 seq(coeff(series((sqrt(1-4*x) - sqrt(1-6*x+x^2))/(2*x) -1/2, x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Nov 01 2019
%t A026773 Rest@CoefficientList[Series[(Sqrt[1-4*x] - Sqrt[1-6*x+x^2])/(2*x) -1/2, {x,0,30}], x] (* _G. C. Greubel_, Nov 01 2019 *)
%o A026773 (PARI) my(x='x+O('x^30)); Vec((sqrt(1-4*x) - sqrt(1-6*x+x^2))/(2*x) -1/2) \\ _G. C. Greubel_, Nov 01 2019
%o A026773 (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (Sqrt(1-4*x) - Sqrt(1-6*x+x^2))/(2*x) -1/2 )); // _G. C. Greubel_, Nov 01 2019
%o A026773 (Sage)
%o A026773 def A026773_list(prec):
%o A026773     P.<x> = PowerSeriesRing(ZZ, prec)
%o A026773     return P((sqrt(1-4*x) - sqrt(1-6*x+x^2))/(2*x) -1/2).list()
%o A026773 a=A026773_list(30); a[1:] # _G. C. Greubel_, Nov 01 2019
%o A026773 (GAP) List([0..30], n-> Sum([0..n], k-> Binomial(n+1,k)*Binomial(n+k+1, n+1)/(k+1) )); # _G. C. Greubel_, Nov 01 2019
%Y A026773 Cf. A026769, A026770, A026771, A026772, A026774, A026775, A026776, A026777, A026778, A026779.
%K A026773 nonn
%O A026773 1,2
%A A026773 _Clark Kimberling_