A360219 a(n) = Sum_{k=0..floor(n/4)} (-1)^k * binomial(n-3*k,k) * binomial(2*(n-3*k),n-3*k).

1, 2, 6, 20, 68, 240, 864, 3152, 11616, 43136, 161152, 604992, 2280416, 8624832, 32714688, 124399488, 474066560, 1810053120, 6922776576, 26517173760, 101710338048, 390603984896, 1501732753408, 5779500226560, 22263437981184, 85835254221824, 331193445626880

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. A157004, A360267, A374599.

Programs

Maple

A360219 := proc(n)
    add((-1)^k*binomial(n-3*k,k)*binomial(2*(n-3*k),n-3*k),k=0..n/3) ;
end proc:
seq(A360219(n),n=0..70) ; # R. J. Mathar, Mar 12 2023

PARI

a(n) = sum(k=0, n\4, (-1)^k*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).