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 A360271 #15 Sep 29 2024 13:17:25
%S A360271 1,1,2,5,13,38,117,373,1222,4085,13877,47766,166229,583893,2067414,
%T A360271 7371093,26440789,95355990,345538389,1257486165,4593933398,
%U A360271 16841578325,61938532181,228454719830,844882459989,3132258655573,11638656376150,43337083401557
%N A360271 a(n) = Sum_{k=0..floor(n/4)} (-1)^k * binomial(n-3*k,k) * Catalan(n-3*k).
%F A360271 G.f.: c(x * (1-x^3)), where c(x) is the g.f. of A000108.
%F A360271 a(n) ~ 2 * sqrt(1-3*r) / (sqrt(Pi) * n^(3/2) * r^n), where r = 0.2541737124933... is the smallest positive root of the equation 1 - 4*r + 4*r^4 = 0. - _Vaclav Kotesovec_, Feb 01 2023
%F A360271 D-finite with recurrence (n+1)*a(n) +2*(-2*n+1)*a(n-1) +(-n-1)*a(n-3) +2*(4*n-11)*a(n-4) +4*(-n+5)*a(n-7)=0. - _R. J. Mathar_, Mar 12 2023
%p A360271 A360271 := proc(n)
%p A360271 add((-1)^k*binomial(n-3*k,k)*A000108(n-3*k),k=0..n/3) ;
%p A360271 end proc:
%p A360271 seq(A360271(n),n=0..70) ; # _R. J. Mathar_, Mar 12 2023
%o A360271 (PARI) a(n) = sum(k=0, n\4, (-1)^k*binomial(n-3*k, k)*binomial(2*(n-3*k), n-3*k)/(n-3*k+1));
%o A360271 (PARI) my(N=30, x='x+O('x^N)); Vec(2/(1+(sqrt(1-4*x*(1-x^3)))))
%Y A360271 Cf. A157003.
%Y A360271 Cf. A000108, A360219, A360272.
%K A360271 nonn
%O A360271 0,3
%A A360271 _Seiichi Manyama_, Jan 31 2023