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 A360059 #25 May 06 2024 22:48:19
%S A360059 1,3,4,3,5,12,6,-13,29,95,-130,-304,895,1050,-5068,-2181,27743,-5481,
%T A360059 -143532,117983,700831,-1074414,-3163138,7872784,12585117,-51587107,
%U A360059 -38040886,312988334,18178883,-1779688404,1013771196,9485832411,-11749675733,-46878057651
%N A360059 a(n) = Sum_{k=0..n} (-1)^k * binomial(n+3*k+3,n-k) * Catalan(k).
%F A360059 a(n) = binomial(n+3,3) - Sum_{k=0..n-1} a(k) * a(n-k-1).
%F A360059 G.f. A(x) satisfies A(x) = 1/(1-x)^4 - x * A(x)^2.
%F A360059 G.f.: 2 / ( (1-x)^4 * (1 + sqrt( 1 + 4*x/(1-x)^4 )) ).
%F A360059 D-finite with recurrence (n+1)*a(n) +(-n-2)*a(n-1) +6*(n-2)*a(n-2) +10*(-n+2)*a(n-3) +5*(n-3)*a(n-4) +(-n+4)*a(n-5)=0. - _R. J. Mathar_, Jan 25 2023
%t A360059 Table[Sum[(-1)^k Binomial[n+3k+3,n-k]CatalanNumber[k],{k,0,n}],{n,0,40}] (* _Harvey P. Dale_, May 06 2024 *)
%o A360059 (PARI) a(n) = sum(k=0, n, (-1)^k*binomial(n+3*k+3, n-k)*binomial(2*k, k)/(k+1));
%o A360059 (PARI) my(N=40, x='x+O('x^N)); Vec(2/((1-x)^4*(1+sqrt(1+4*x/(1-x)^4))))
%Y A360059 Cf. A360058, A360060.
%Y A360059 Cf. A000108, A358518, A360050.
%K A360059 sign
%O A360059 0,2
%A A360059 _Seiichi Manyama_, Jan 23 2023