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%I #23 Jan 16 2024 01:38:23
%S 1,3,27,270,2835,30618,336798,3752892,42220035,478493730,5454828522,
%T 62482581252,718549684398,8290957896900,95938227092700,
%U 1112883434275320,12937269923450595,150681143814306930,1757946677833580850,20540219077844997300,240320563210786468410
%N a(n) = (0^n + 3^n*binomial(2*n,n))/2.
%H Vincenzo Librandi, <a href="/A098401/b098401.txt">Table of n, a(n) for n = 0..900</a>
%F a(n+1) = 3*A098399(n).
%F G.f.: 6*x/(sqrt(1-12*x)*(1-sqrt(1-12*x)).
%F n*a(n) - 6*(2*n-1)*a(n-1) = 0. - _R. J. Mathar_, Nov 24 2012
%F From _Amiram Eldar_, Jan 16 2024: (Start)
%F Sum_{n>=0} 1/a(n) = 13/11 + 24*arcsin(1/(2*sqrt(3)))/(11*sqrt(11)).
%F Sum_{n>=0} (-1)^n/a(n) = 11/13 - 24*arcsinh(1/(2*sqrt(3)))/(13*sqrt(13)). (End)
%t CoefficientList[Series[(6x)/(Sqrt[1-12x](1-Sqrt[1-12x])),{x,0,30}],x] (* _Harvey P. Dale_, Nov 29 2023 *)
%t Table[(3^n*Binomial[2*n,n] +Boole[n==0])/2, {n,0,40}] (* _G. C. Greubel_, Dec 27 2023 *)
%o (Magma) [(0^n + 3^n * Binomial(2*n, n))/2: n in [ 0..20]]; // _Vincenzo Librandi_, Nov 24 2012
%o (SageMath) [(3^n*binomial(2*n,n) + int(n==0))/2 for n in range(41)] # _G. C. Greubel_, Dec 27 2023
%Y Cf. A069723, A069720, A098399, A098400, A098402.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Sep 06 2004
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