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 A143669 #23 Sep 08 2022 08:45:37
%S A143669 1,1,4,35,506,10472,285384,9706503,397089550,19022318084,
%T A143669 1045659267016,64924369564353,4496010926381352,343688726144945040,
%U A143669 28753733905585301136,2613784129155164386575,256569498889138342791510,27050758656206146528056236
%N A143669 a(n) = binomial((n+1)^2, n) / (n+1)^2.
%C A143669 From _Peter Bala_, Dec 02 2015: (Start)
%C A143669 Let x = p/q be a positive rational in reduced form with p,q > 0. Define Cat(x) = 1/(2*p + q)*binomial(2*p + q, p). Then Cat(n) = Catalan(n). This sequence is Cat(n/(n^2 + 1)). Cf. A135862.
%C A143669 See Armstrong et al. for combinatorial interpretations of these generalized Catalan number sequences. (End)
%H A143669 Seiichi Manyama, <a href="/A143669/b143669.txt">Table of n, a(n) for n = 0..339</a>
%H A143669 D. Armstrong, B. Rhoades, and N. Williams, <a href="http://arxiv.org/abs/1305.7286">Rational associahedra and noncrossing partitions</a> arxiv:1305.7286v1 [math.CO], 2013.
%t A143669 Table[Binomial[(n + 1)^2, n]/(n + 1)^2, {n, 0, 30}] (* _Vincenzo Librandi_, Dec 09 2015 *)
%o A143669 (PARI) a(n)=binomial((n+1)^2,n)/(n+1)^2
%o A143669 (Magma) [Binomial((n+1)^2, n) / (n+1)^2: n in [0..20]]; // _Vincenzo Librandi_, Dec 09 2015
%Y A143669 Cf. A295765, A135862, A182316.
%K A143669 nonn,easy
%O A143669 0,3
%A A143669 _Paul D. Hanna_, Aug 28 2008