A360150 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-k,n-3*k).

1, 2, 6, 21, 77, 288, 1090, 4159, 15964, 61557, 238221, 924597, 3597290, 14024341, 54770176, 214218966, 838959762, 3289471537, 12910910288, 50720828034, 199422778415, 784672001097, 3089564308849, 12172411084432, 47984843655991, 189260578353602

Offset: 0

Author

Seiichi Manyama, Jan 28 2023

Keywords

• nonn

Crossrefs

Cf. A105872, A144904, A360151, A360152, A360153, A360168.

Cf. A000108.

Programs

Maple

A360150 := proc(n)
    add(binomial(2*n-k,n-3*k),k=0..n/3) ;
end proc:
seq(A360150(n),n=0..70) ; # R. J. Mathar, Mar 12 2023

Mathematica

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

PARI

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

PARI

my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1-x^3*(2/(1+sqrt(1-4*x)))^5)))

Formula

G.f.: 1 / ( sqrt(1-4*x) * (1 - x^3 * c(x)^5) ), where c(x) is the g.f. of A000108.

a(n) ~ 2^(2*n+1) / sqrt(Pi*n). - Vaclav Kotesovec, Jan 28 2023

D-finite with recurrence n*a(n) +2*(-7*n+6)*a(n-1) +2*(36*n-61)*a(n-2) +4*(-41*n+103)*a(n-3) +(161*n-530)*a(n-4) +(-71*n+278)*a(n-5) +6*(2*n-9)*a(n-6)=0. - R. J. Mathar, Mar 12 2023

a(n) = [x^n] 1/(((1-x)^2-x^3) * (1-x)^(n-1)). - Seiichi Manyama, Apr 09 2024