A393250 a(n) = Sum_{k=0..floor(n/6)} binomial(2*k,k) * binomial(n-3*k,3*k).

1, 1, 1, 1, 1, 1, 3, 9, 21, 41, 71, 113, 175, 283, 499, 945, 1833, 3501, 6475, 11617, 20479, 36063, 64287, 116577, 214215, 395583, 728673, 1333883, 2427239, 4403243, 7991759, 14548745, 26584661, 48714993, 89370903, 163903509, 300276435, 549583299, 1005524031

Offset

0,7

Links

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

Formula

G.f.: 1/sqrt((1-x)^2 - 4*x^6/(1-x)).

D-finite with recurrence: (10 + 4*n)*a(n) + (-4*n - 16)*a(n + 1) + (n + 4)*a(n + 3) + (-19 - 4*n)*a(n + 4) + (33 + 6*n)*a(n + 5) + (-25 - 4*n)*a(n + 6) + (n + 7)*a(n + 7) = 0. - Robert Israel, Feb 09 2026

Maple

f:= gfun:-rectoproc({(10 + 4*n)*a(n) + (-4*n - 16)*a(n + 1) + (n + 4)*a(n + 3) + (-19 - 4*n)*a(n + 4) + (33 + 6*n)*a(n + 5) + (-25 - 4*n)*a(n + 6) + (n + 7)*a(n + 7), a(0) = 1, a(1) = 1, a(2) = 1, a(3) = 1, a(4) = 1, a(5) = 1, a(6) = 3}, a(n), remember):
map(f, [$0..40]); # Robert Israel, Feb 09 2026

Mathematica

Table[Sum[Binomial[2*k, k]*Binomial[n-3*k, 3*k], {k, 0, Floor[n/6]}], {n, 0, 45}] (* Vincenzo Librandi, Feb 10 2026 *)

Prog

(PARI) a(n) = sum(k=0, n\6, binomial(2*k, k)*binomial(n-3*k, 3*k));
(Magma) [&+[Binomial(2*k, k)* Binomial(n-3*k, 3*k) : k in [0..Floor(n/6)]] : n in [0..43] ]; // Vincenzo Librandi, Feb 10 2026

Crossrefs

Cf. A393251.

Sequence in context: A059774 A064999 A100135 * A393247 A024173 A396151

Adjacent sequences: A393247 A393248 A393249 * A393251 A393252 A393253

Keyword

nonn,easy

Author

Seiichi Manyama, Feb 07 2026

Status

approved