A360290 a(n) = Sum_{k=0..floor(n/2)} binomial(n-1-k,k) * binomial(2*n-4*k,n-2*k).

1, 2, 6, 22, 82, 314, 1222, 4814, 19138, 76626, 308550, 1248230, 5069266, 20654602, 84392838, 345659166, 1418769154, 5834283298, 24031706246, 99134911542, 409495076050, 1693539077210, 7011618614342, 29058701620974, 120540377731266, 500443750830962

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Formula

G.f.: 1 / sqrt(1-4*x/(1-x^2)).

n*a(n) = 2*(2*n-1)*a(n-1) + 2*(n-2)*a(n-2) - 2*(2*n-7)*a(n-3) - (n-4)*a(n-4).

a(n) ~ phi^(3*n) / (5^(1/4) * sqrt(Pi*n/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Feb 02 2023

a(n) = A383573(n) - A383573(n-2). - Seiichi Manyama, May 01 2025

Mathematica

Table[Sum[Binomial[n-1-k, k]* Binomial[2*n-4*k, n-2*k], {k, 0, Floor[n/2]}], {n, 0, 35}] (* Vincenzo Librandi, May 04 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\2, binomial(n-1-k, k)*binomial(2*n-4*k, n-2*k));
(PARI) my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1-x^2)))
(Magma) [&+[Binomial(n-1-k, k) * Binomial(2*n-4*k, n-2*k): k in [0..Floor(n div 2)]]: n in [0..30]]; // Vincenzo Librandi, May 04 2025

Crossrefs

Cf. A085362, A360291, A360292.

Cf. A360185, A360293, A383573.

Sequence in context: A150230 A360266 A382189 * A191755 A150231 A150232

Adjacent sequences: A360287 A360288 A360289 * A360291 A360292 A360293

Keyword

nonn

Author

Seiichi Manyama, Feb 01 2023

Status

approved