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 A360319 #18 Aug 22 2025 11:02:21
%S A360319 1,2,14,100,726,5340,39692,297544,2245990,17050796,130061412,
%T A360319 996078456,7654571772,58995989400,455857911768,3530234227344,
%U A360319 27392392806534,212918339726028,1657570714812020,12922254685161112,100867892292766612
%N A360319 a(n) = Sum_{k=0..n} 4^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).
%F A360319 G.f.: sqrt( (1-4*x)/(1-8*x) ).
%F A360319 n*a(n) = 2*(6*n-5)*a(n-1) - 32*(n-2)*a(n-2).
%F A360319 Sum_{i=0..n} Sum_{j=0..i} (1/4)^i * a(j) * a(i-j) = 2^n.
%F A360319 a(n) ~ 2^(3*n - 1/2) / sqrt(Pi*n). - _Vaclav Kotesovec_, Feb 04 2023
%F A360319 From _Seiichi Manyama_, Aug 22 2025: (Start)
%F A360319 a(n) = Sum_{k=0..n} 2^k * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
%F A360319 a(n) = Sum_{k=0..n} (-1)^k * 8^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n-1,n-k). (End)
%o A360319 (PARI) a(n) = sum(k=0, n, 4^(n-k)*binomial(n-1, n-k)*binomial(2*k, k));
%o A360319 (PARI) my(N=30, x='x+O('x^N)); Vec(sqrt((1-4*x)/(1-8*x)))
%Y A360319 Cf. A063886, A085362, A360317, A360318, A360321, A360322.
%K A360319 nonn,easy,changed
%O A360319 0,2
%A A360319 _Seiichi Manyama_, Feb 03 2023