A383583 a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k-1,k) * binomial(k,n-3*k).

1, 0, 0, 0, 1, 0, 0, 2, 3, 0, 3, 12, 10, 4, 30, 60, 40, 60, 210, 286, 231, 560, 1267, 1428, 1722, 4208, 7182, 8064, 13275, 28080, 40656, 51754, 97020, 176088, 240251, 355872, 667810, 1081092, 1506648, 2475616, 4401696, 6693492, 9904752, 16950662, 28359201

Offset: 0

Author

Seiichi Manyama, May 01 2025

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A005717, A383584.

Cf. A383571.

Programs

Magma

[&+[Binomial(n-2*k-1, k) * Binomial(k, n-3*k): k in [0..Floor(n div 3)]]: n in [0..45]]; // Vincenzo Librandi, May 03 2025

Mathematica

Table[Sum[Binomial[n-2*k-1,k]* Binomial[k,n-3*k],{k,0,Floor[n/3]}],{n,0,45}] (* Vincenzo Librandi, May 03 2025 *)

PARI

a(n) = sum(k=0, n\3, binomial(n-2*k-1, k)*binomial(k, n-3*k));

Formula

G.f.: (1/2) * ( 1 + 1/sqrt(1 - 4*x^4/(1-x^3)^2) ).

a(n) ~ phi^(n-1) / (2 * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, May 01 2025