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 A360290 #22 May 04 2025 03:20:33
%S A360290 1,2,6,22,82,314,1222,4814,19138,76626,308550,1248230,5069266,
%T A360290 20654602,84392838,345659166,1418769154,5834283298,24031706246,
%U A360290 99134911542,409495076050,1693539077210,7011618614342,29058701620974,120540377731266,500443750830962
%N A360290 a(n) = Sum_{k=0..floor(n/2)} binomial(n-1-k,k) * binomial(2*n-4*k,n-2*k).
%H A360290 Vincenzo Librandi, <a href="/A360290/b360290.txt">Table of n, a(n) for n = 0..400</a>
%F A360290 G.f.: 1 / sqrt(1-4*x/(1-x^2)).
%F A360290 n*a(n) = 2*(2*n-1)*a(n-1) + 2*(n-2)*a(n-2) - 2*(2*n-7)*a(n-3) - (n-4)*a(n-4).
%F A360290 a(n) ~ phi^(3*n) / (5^(1/4) * sqrt(Pi*n/2)), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Feb 02 2023
%F A360290 a(n) = A383573(n) - A383573(n-2). - _Seiichi Manyama_, May 01 2025
%t A360290 Table[Sum[Binomial[n-1-k,k]* Binomial[2*n-4*k, n-2*k],{k,0,Floor[n/2]}],{n,0,35}] (* _Vincenzo Librandi_, May 04 2025 *)
%o A360290 (PARI) a(n) = sum(k=0, n\2, binomial(n-1-k, k)*binomial(2*n-4*k, n-2*k));
%o A360290 (PARI) my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1-x^2)))
%o A360290 (Magma) [&+[Binomial(n-1-k, k) * Binomial(2*n-4*k, n-2*k): k in [0..Floor(n div 2)]]: n in [0..30]]; // _Vincenzo Librandi_, May 04 2025
%Y A360290 Cf. A085362, A360291, A360292.
%Y A360290 Cf. A360185, A360293, A383573.
%K A360290 nonn
%O A360290 0,2
%A A360290 _Seiichi Manyama_, Feb 01 2023