A393274 a(n) = Sum_{k=0..floor(n/3)} binomial(2*k,k) * binomial(n-1,n-3*k).

1, 0, 0, 2, 6, 12, 26, 66, 168, 412, 1008, 2502, 6252, 15624, 39078, 97986, 246258, 619848, 1562204, 3942306, 9960750, 25194316, 63787206, 161641554, 409951990, 1040508996, 2642812140, 6716956610, 17082258372, 43467728976, 110667864798, 281899652166, 718410766536

Offset

0,4

Links

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

Formula

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

D-finite with recurrence: 5*n*a(n) + (-14 - 8*n)*a(n + 1) + (6*n + 12)*a(n + 2) + (-4*n - 12)*a(n + 3) + (n + 4)*a(n + 4) = 0. - Robert Israel, Mar 08 2026

Maple

f:= gfun:-rectoproc({5*n*a(n) + (-14 - 8*n)*a(n + 1) + (6*n + 12)*a(n + 2) + (-4*n - 12)*a(n + 3) + (n + 4)*a(n + 4), a(0) = 1, a(1) = 0, a(2) = 0, a(3) = 2}, a(n), remember):
map(f, [$0..45]); # Robert Israel, Mar 08 2026

Mathematica

Table[Sum[Binomial[2*k, k]*Binomial[n-1, n-3*k], {k, 0, Floor[n/3]}], {n, 0, 45}] (* Vincenzo Librandi, Feb 10 2026 *)

Prog

(PARI) a(n) = sum(k=0, n\3, binomial(2*k, k)*binomial(n-1, n-3*k));
(Magma) [&+[Binomial(2*k, k)* Binomial(n-1, n-3*k) : k in [0..Floor(n/3)]] : n in [0..43] ]; // Vincenzo Librandi, Feb 10 2026

Crossrefs

Partial sums are A393244.

Cf. A025178, A085362, A393275.

Sequence in context: A380176 A052971 A364423 * A289443 A029863 A091919

Adjacent sequences: A393271 A393272 A393273 * A393275 A393276 A393277

Keyword

nonn,easy

Author

Seiichi Manyama, Feb 08 2026

Status

approved