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 A377194 #13 May 09 2025 00:53:53
%S A377194 1,6,42,266,1650,10032,60202,357744,2109882,12369280,72163560,
%T A377194 419315340,2428226530,14021002860,80757350040,464127134636,
%U A377194 2662303793226,15245389224492,87168383093576,497721319382220,2838427001118456,16168991846946656,92012074475132892
%N A377194 Expansion of 1/(1 - 4*x*(1+x)^2)^(3/2).
%F A377194 a(0) = 1, a(1) = 6, a(2) = 42; a(n) = (2*(2*n+1)*a(n-1) + 8*(n+1)*a(n-2) + 2*(2*n+3)*a(n-3))/n.
%F A377194 a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(2*k,n-k).
%t A377194 a[n_]:=Sum[(2*k+1)Binomial[2*k,k]Binomial[2k,n-k],{k,0,n}]; Array[a,23,0] (* _Stefano Spezia_, May 08 2025 *)
%o A377194 (PARI) a(n) = sum(k=0, n, (2*k+1)*binomial(2*k, k)*binomial(2*k, n-k));
%Y A377194 Cf. A137635, A377195, A377196.
%K A377194 nonn
%O A377194 0,2
%A A377194 _Seiichi Manyama_, Oct 19 2024