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 A286033 #26 Jan 21 2025 10:40:35
%S A286033 0,3,5,21,69,253,923,3433,12869,48621,184755,705433,2704155,10400601,
%T A286033 40116599,155117521,601080389,2333606221,9075135299,35345263801,
%U A286033 137846528819,538257874441,2104098963719,8233430727601,32247603683099,126410606437753,495918532948103
%N A286033 a(n) = binomial(2*n-2, n-1) + (-1)^n.
%C A286033 An odd prime p divides a((p+1)/2) which gives A163210.
%H A286033 G. C. Greubel, <a href="/A286033/b286033.txt">Table of n, a(n) for n = 1..1000</a>
%F A286033 a(n) = A000984(n-1) + A033999(n). - _David A. Corneth_, May 13 2017
%F A286033 G.f.: -1 + x/sqrt(1 - 4*x) + 1/(1 + x). - _Ilya Gutkovskiy_, May 13 2017
%F A286033 D-finite with recurrence: -(n-1)*a(n) +2*(n-1)*a(n-1) +(7*n-17)*a(n-2) +2*(2*n-7)*a(n-3)=0. - _R. J. Mathar_, Jan 27 2020
%F A286033 a(n) = Sum_{k=1..n} (-1)^(k-1)*binomial(2*n, n-k)*A000032(2*k). - _Vladimir Kruchinin_, Jan 18 2025
%p A286033 a := n -> binomial(2*n-2, n-1) + (-1)^n: seq(a(n), n=1..27);
%t A286033 a[n_] := Binomial[2n-2, n-1] + (-1)^n; a[Range[1,27]]
%o A286033 (PARI) a(n) = binomial(2*n-2, n-1) + (-1)^n \\ _David A. Corneth_, May 13 2017
%o A286033 (Magma) [Binomial(2*n-2, n-1) + (-1)^n: n in [1..30]]; // _G. C. Greubel_, Jul 14 2024
%o A286033 (SageMath)
%o A286033 def A286033(n): return binomial(2*n-2, n-1) + (-1)^n
%o A286033 [A286033(n) for n in range(1,31)] # _G. C. Greubel_, Jul 14 2024
%o A286033 (Maxima) a(n):=-sum((-1)^k*binomial(2*n,n-k)*(fib(2*k+1)+fib(2*k-1)),k,1,n); /* _Vladimir Kruchinin_, Jan 18 2025 */
%Y A286033 Cf. A000984, A033999, A163210, A178904, A219539, A000032.
%K A286033 nonn,easy
%O A286033 1,2
%A A286033 _Peter Luschny_, May 13 2017