A360151 - OEIS

%I #12 Mar 12 2023 11:20:44

%S 1,2,6,21,74,267,981,3648,13690,51744,196699,751237,2880345,11080081,

%T 42743148,165291569,640563158,2487083484,9672626600,37674470433,

%U 146937686295,573781535775,2243050091905,8777451670102,34379401083017,134770951530840

%N a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-4*k,n-3*k).

%F G.f.: 1 / ( sqrt(1-4*x) * (1 - x^3 * c(x)^2) ), where c(x) is the g.f. of A000108.

%F a(n) ~ 2^(2*n+4) / (15*sqrt(Pi*n)). - _Vaclav Kotesovec_, Jan 28 2023

%F D-finite with recurrence +2*n*a(n) +(-11*n+6)*a(n-1) +(19*n-24)*a(n-2) +2*(-16*n+33)*a(n-3) +2*(11*n-36)*a(n-4) +(-25*n+78)*a(n-5) +6*(n-3)*a(n-6) +4*(-2*n+9)*a(n-7)=0. - _R. J. Mathar_, Mar 12 2023

%p A360151 := proc(n)

%p     add(binomial(2*n-4*k,n-3*k),k=0..n/3) ;

%p end proc:

%p seq(A360151(n),n=0..70) ; # _R. J. Mathar_, Mar 12 2023

%t a[n_] := Sum[Binomial[2*n - 4*k, n - 3*k], {k, 0, Floor[n/3]}]; Array[a, 26, 0] (* _Amiram Eldar_, Jan 28 2023 *)

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

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

%Y Cf. A105872, A144904, A360150, A360152, A360153.

%Y Cf. A000108.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jan 28 2023