OFFSET
0,4
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: 1/sqrt(1 - 4 * x^3 * (1+x)).
a(n) = Sum_{k=0..floor(n/3)} binomial(2*k,k) * binomial(k,n-3*k).
From Vaclav Kotesovec, Mar 23 2023: (Start)
Recurrence: n*a(n) = 2*(2*n-3)*a(n-3) + 4*(n-2)*a(n-4).
a(n) ~ sqrt(c) * d^n / sqrt(Pi*n), where d = 1.835086681639635368143322042736678753... is the positive real root of the equation d^4 - 4*d - 4 = 0 and c = 0.2982650309662120181812121016104223... is the largest real root of the equation 1 - 20*c + 132*c^2 - 364*c^3 + 364*c^4 = 0. (End)
MATHEMATICA
Table[Sum[Binomial[2*k, k] * Binomial[k, n-3*k], {k, 0, n/3}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 23 2023 *)
PROG
(PARI) a(n) = sum(k=0, n\3, binomial(2*k, k)*binomial(k, n-3*k));
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Mar 22 2023
STATUS
approved