A360272 - OEIS

%I #16 Sep 29 2024 13:17:20

%S 1,1,2,5,15,46,147,485,1642,5669,19883,70646,253755,919925,3361546,

%T 12368661,45786219,170400470,637200555,2392962645,9021255722,

%U 34128098389,129519490219,492966689110,1881289209003,7197100511317,27595769836714,106032318322517

%N a(n) = Sum_{k=0..floor(n/4)} binomial(n-3*k,k) * Catalan(n-3*k).

%F G.f.: c(x * (1+x^3)), where c(x) is the g.f. of A000108.

%F a(n) ~ 2 * sqrt(1-3*r) / (sqrt(Pi) * n^(3/2) * r^n), where r = 0.2463187933841... is the smallest positive root of the equation1 1 - 4*r - 4*r^4 = 0. - _Vaclav Kotesovec_, Feb 01 2023

%F 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 A360272 := proc(n)

%p     add(binomial(n-3*k,k)*A000108(n-3*k),k=0..n/3) ;

%p end proc:

%p seq(A360272(n),n=0..70) ; # _R. J. Mathar_, Mar 12 2023

%o (PARI) a(n) = sum(k=0, n\4, binomial(n-3*k, k)*binomial(2*(n-3*k), n-3*k)/(n-3*k+1));

%o (PARI) my(N=30, x='x+O('x^N)); Vec(2/(1+(sqrt(1-4*x*(1+x^3)))))

%Y Cf. A052709, A125305.

%Y Cf. A000108, A360267, A360271, A360274.

%K nonn,changed

%O 0,3

%A _Seiichi Manyama_, Jan 31 2023