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 A274104 #24 Sep 03 2024 00:42:12
%S A274104 2,7,23,78,274,988,3628,13495,50675,191673,729145,2786655,10691111,
%T A274104 41150011,158825371,614483086,2382366586,9253540456,36001307656,
%U A274104 140269835866,547245301906,2137552658206,8358366985726,32715599554876,128168506456852,502538379368656,1971926625140816
%N A274104 a(n) = Sum_{k=0..n} (3*k+2)*Catalan(k).
%H A274104 Vincenzo Librandi, <a href="/A274104/b274104.txt">Table of n, a(n) for n = 0..200</a>
%H A274104 Moa Apagodu and Doron Zeilberger, <a href="http://arxiv.org/abs/1606.03351">Using the "Freshman's Dream" to Prove Combinatorial Congruences</a>, arXiv:1606.03351 [math.CO], 2016. Also Amer. Math. Monthly. 124 (2017), 597-608.
%F A274104 D-finite with recurrence: (n+1)*a(n) - (3*n+5)*a(n-1) - 2*(3*n-8)*a(n-2) + 4*(2*n-3)*a(n-3) = 0. - _R. J. Mathar_, Jun 15 2016
%F A274104 G.f.: (1 + 2*x - sqrt(1-4*x))/(2*x*(1-x)*sqrt(1-4*x)). - _Ilya Gutkovskiy_, Jun 15 2016
%F A274104 a(n) = A014137(n+1) + (n+1)*A000108(n+1) - 1. - _G. C. Greubel_, Jun 30 2024
%F A274104 From _Mélika Tebni_, Sep 02 2024: (Start)
%F A274104 a(n) = A006134(n) + A006134(n+1)/2 - 1/2.
%F A274104 E.g.f.: exp(2*x)*(5*BesselI(0, 2*x)/2 + BesselI(1, 2*x)) + exp(x)/2*(3*Integral_{x=-oo..oo} BesselI(0,2*x)*exp(x) dx - 1). (End)
%t A274104 CoefficientList[Series[(1 +2 x -Sqrt[1-4 x])/(2 x Sqrt[1-4 x] (1-x)), {x, 0, 50}], x] (* _Vincenzo Librandi_, Aug 18 2016 *)
%o A274104 (Magma) [(&+[(3*k+2)*Catalan(k): k in [0..n]]): n in [0..40]]; // _G. C. Greubel_, Jun 30 2024
%o A274104 (SageMath) [sum((3*k+2)*catalan_number(k) for k in range(n+1)) for n in range(41)] # _G. C. Greubel_, Jun 30 2024
%Y A274104 Partial sums of A051960.
%Y A274104 Cf. A000108, A006134, A014137.
%K A274104 nonn
%O A274104 0,1
%A A274104 _N. J. A. Sloane_, Jun 13 2016