OFFSET
0,2
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
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
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 *)
PROG
(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)))
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Jan 31 2023
STATUS
approved