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 A277871 #25 Apr 10 2017 23:00:24
%S A277871 1,4,16,66,279,1203,5275,23474,105853,483108,2229253,10390691,
%T A277871 48879588,231879456,1108473015,5335987930,25849521109,125945214309,
%U A277871 616833862018,3035286848660,14999774773110,74413424196360,370463714276625,1850251796668899
%N A277871 a(n) = Sum_{i=0..n+1} binomial(2*n-i,n-i+1)*Catalan(i).
%C A277871 T(2*n+1,n) is diagonal of triangle A125177.
%H A277871 Seiichi Manyama, <a href="/A277871/b277871.txt">Table of n, a(n) for n = 0..1381</a>
%F A277871 G.f.: (4*x*(1-sqrt(1-2*(1-sqrt(1-4*x)))))/(1-sqrt(1-4*x))^3/sqrt(1-4*x)-1/x.
%F A277871 a(n) ~ 2^(4*n+1/2) / (sqrt(Pi) * n^(3/2) * 3^(n-3/2)). - _Vaclav Kotesovec_, Nov 05 2016
%t A277871 Table[Sum[Binomial[2*i, i]*Binomial[2*n-i, n-i+1]/(i+1), {i, 0, n+1}], {n, 0, 20}] (* _Vaclav Kotesovec_, Nov 05 2016 *)
%o A277871 (Maxima)
%o A277871 a(n):=sum((binomial(2*i,i)*binomial(2*n-i,n-i+1))/(i+1),i,0,n+1);
%o A277871 (PARI) x='x+O('x^50); Vec((4*x*(1-sqrt(1-2*(1-sqrt(1-4*x)))))/(1-sqrt(1-4*x))^3/sqrt(1-4*x)-1/x) \\ _G. C. Greubel_, Apr 09 2017
%Y A277871 Cf. A000108, A125177.
%K A277871 nonn
%O A277871 0,2
%A A277871 _Vladimir Kruchinin_, Nov 02 2016