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 A358518 #52 Mar 11 2023 08:09:22
%S A358518 1,5,20,85,405,2116,11766,68237,407789,2492553,15506942,97859544,
%T A358518 624880895,4029896310,26209648212,171711104853,1132143259711,
%U A358518 7506530891217,50019287312324,334784759816729,2249720564735567,15172573979205166,102662981205576494
%N A358518 a(n) = Sum_{k=0..n} binomial(n+3*k+3,n-k) * Catalan(k).
%F A358518 a(n) = binomial(n+3,3) + Sum_{k=0..n-1} a(k) * a(n-k-1).
%F A358518 G.f. A(x) satisfies A(x) = 1/(1-x)^4 + x * A(x)^2.
%F A358518 G.f.: 2 / ( (1-x)^4 * (1 + sqrt( 1 - 4*x/(1-x)^4 )) ).
%F A358518 D-finite with recurrence (n+1)*a(n) +(-9*n+2)*a(n-1) +2*(7*n-4)*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
%o A358518 (PARI) a(n) = sum(k=0, n, binomial(n+3*k+3, n-k)*binomial(2*k, k)/(k+1));
%o A358518 (PARI) my(N=30, x='x+O('x^N)); Vec(2/((1-x)^4*(1+sqrt(1-4*x/(1-x)^4))))
%Y A358518 Cf. A086616, A162481, A360057.
%Y A358518 Cf. A000108, A360046.
%K A358518 nonn
%O A358518 0,2
%A A358518 _Seiichi Manyama_, Jan 23 2023