A360267 a(n) = Sum_{k=0..floor(n/4)} binomial(n-3*k,k) * binomial(2*(n-3*k),n-3*k).

1, 2, 6, 20, 72, 264, 984, 3712, 14136, 54224, 209200, 810912, 3155616, 12320512, 48239232, 189336192, 744722400, 2934759360, 11584470336, 45796087680, 181285742592, 718498695424, 2850802065152, 11322567705600, 45011437903104, 179088911779328

Offset: 0

Author

Seiichi Manyama, Jan 31 2023

Keywords

• nonn

Comments

Diagonal of rational function 1/(1 - (x + y + x^4*y^3)). - Seiichi Manyama, Mar 23 2023

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A006139, A360266.

Cf. A360219, A374599.

Programs

Mathematica

Table[Sum[Binomial[n-3k,k]Binomial[2(n-3k),n-3k],{k,0,Floor[n/3]}],{n,0,30}] (* Harvey P. Dale, May 27 2024 *)

PARI

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

PARI

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

Formula

G.f.: 1/sqrt(1 - 4*x*(1 + x^3)).

n*a(n) = 2*(2*n-1)*a(n-1) + 2*(2*n-4)*a(n-4).

a(n) ~ 1 / (2*sqrt((1 - 3*r)*Pi*n) * r^n), where r = 0.2463187933841190115229... is the positive real root of the equation -1 + 4*r + 4*r^4 = 0. - Vaclav Kotesovec, Mar 23 2023