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 A360045 #16 Feb 06 2025 13:23:26
%S A360045 1,3,6,11,21,42,86,180,387,852,1908,4329,9933,23013,53748,126386,
%T A360045 298953,710847,1698086,4073181,9806565,23689555,57401322,139475190,
%U A360045 339767545,829638396,2030206248,4978136133,12229451631,30095772966,74184390468,183139941384
%N A360045 a(n) = Sum_{k=0..floor(n/3)} binomial(n+2,3*k+2) * Catalan(k).
%F A360045 a(n) = binomial(n+2,2) + Sum_{k=0..n-3} a(k) * a(n-k-3).
%F A360045 G.f. A(x) satisfies: A(x) = 1/(1-x)^3 + x^3 * A(x)^2.
%F A360045 G.f.: 2 / ( (1-x) * ((1-x)^2 + sqrt((1-x)^4 - 4*x^3*(1-x))) ).
%F A360045 D-finite with recurrence (n+3)*a(n) +4*(-n-2)*a(n-1) +6*(n+1)*a(n-2) +2*(-4*n+3)*a(n-3) +5*(n-1)*a(n-4)=0. - _R. J. Mathar_, Jan 25 2023
%o A360045 (PARI) a(n) = sum(k=0,n\3, binomial(n+2, 3*k+2)*binomial(2*k, k)/(k+1));
%o A360045 (PARI) my(N=40, x='x+O('x^N)); Vec(2/((1-x)*((1-x)^2+sqrt((1-x)^4-4*x^3*(1-x)))))
%Y A360045 Cf. A086615, A360046, A360047.
%Y A360045 Cf. A000108.
%K A360045 nonn
%O A360045 0,2
%A A360045 _Seiichi Manyama_, Jan 23 2023