This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A098401 #31 Aug 30 2024 02:55:41 %S A098401 1,3,27,270,2835,30618,336798,3752892,42220035,478493730,5454828522, %T A098401 62482581252,718549684398,8290957896900,95938227092700, %U A098401 1112883434275320,12937269923450595,150681143814306930,1757946677833580850,20540219077844997300,240320563210786468410 %N A098401 a(n) = (0^n + 3^n*binomial(2*n,n))/2. %H A098401 Vincenzo Librandi, <a href="/A098401/b098401.txt">Table of n, a(n) for n = 0..900</a> %F A098401 a(n+1) = 3*A098399(n). %F A098401 G.f.: 6*x/(sqrt(1-12*x)*(1-sqrt(1-12*x))). %F A098401 n*a(n) - 6*(2*n-1)*a(n-1) = 0. - _R. J. Mathar_, Nov 24 2012 %F A098401 From _Amiram Eldar_, Jan 16 2024: (Start) %F A098401 Sum_{n>=0} 1/a(n) = 13/11 + 24*arcsin(1/(2*sqrt(3)))/(11*sqrt(11)). %F A098401 Sum_{n>=0} (-1)^n/a(n) = 11/13 - 24*arcsinh(1/(2*sqrt(3)))/(13*sqrt(13)). (End) %F A098401 a(n) = 3^n*hypergeom([-n, -n + 1], [1], 1). - _Detlef Meya_, May 21 2024 %t A098401 CoefficientList[Series[(6x)/(Sqrt[1-12x](1-Sqrt[1-12x])),{x,0,30}],x] (* _Harvey P. Dale_, Nov 29 2023 *) %t A098401 Table[(3^n*Binomial[2*n,n] +Boole[n==0])/2, {n,0,40}] (* _G. C. Greubel_, Dec 27 2023 *) %t A098401 a[n_] := 3^n*HypergeometricPFQ[{-n, -n + 1}, {1}, 1]; Flatten[Table[a[n], {n,0,20}]] (* _Detlef Meya_, May 21 2024 *) %o A098401 (Magma) [(0^n + 3^n * Binomial(2*n, n))/2: n in [ 0..20]]; // _Vincenzo Librandi_, Nov 24 2012 %o A098401 (SageMath) [(3^n*binomial(2*n,n) + int(n==0))/2 for n in range(41)] # _G. C. Greubel_, Dec 27 2023 %Y A098401 Cf. A069723, A069720, A098399, A098400, A098402. %K A098401 easy,nonn %O A098401 0,2 %A A098401 _Paul Barry_, Sep 06 2004