A393249 a(n) = Sum_{k=0..floor(n/5)} binomial(2*k,k) * binomial(n-4*k,k).

1, 1, 1, 1, 1, 3, 5, 7, 9, 11, 19, 33, 53, 79, 111, 169, 273, 443, 699, 1061, 1619, 2533, 4033, 6419, 10061, 15651, 24455, 38565, 61151, 96713, 152257, 239395, 377369, 596999, 945555, 1495985, 2363931, 3735965, 5912477, 9368647, 14849803, 23530337, 37278221, 59080035

Offset

0,6

Links

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

Formula

G.f.: 1/sqrt((1-x) * (1-x-4*x^5)).

D-finite with recurrence: (12 + 4*n)*a(n) + (-14 - 4*n)*a(n + 1) + (n + 5)*a(n + 4) + (-11 - 2*n)*a(n + 5) + (n + 6)*a(n + 6) = 0. - Robert Israel, Feb 09 2026

Maple

f:= gfun:-rectoproc({(12 + 4*n)*a(n) + (-14 - 4*n)*a(n + 1) + (n + 5)*a(n + 4) + (-11 - 2*n)*a(n + 5) + (n + 6)*a(n + 6), a(0) = 1, a(1) = 1, a(2) = 1, a(3) = 1, a(4) = 1, a(5) = 3}, a(n), remember):
map(f, [$0..40]); # Robert Israel, Feb 09 2026

Mathematica

Table[Sum[Binomial[2*k, k]*Binomial[n-4*k, k], {k, 0, Floor[n/5]}], {n, 0, 45}] (* Vincenzo Librandi, Feb 08 2026 *)

Prog

(PARI) a(n) = sum(k=0, n\5, binomial(2*k, k)*binomial(n-4*k, k));
(Magma) [&+[Binomial(2*k, k)* Binomial(n-4*k, k) : k in [0..Floor(n/5)]] : n in [0..43] ]; // Vincenzo Librandi, Feb 08 2026

Crossrefs

Cf. A026375, A026569, A217615, A383355.

Cf. A387599, A393247.

Sequence in context: A033032 A103148 A211138 * A178995 A346604 A071240

Adjacent sequences: A393246 A393247 A393248 * A393250 A393251 A393252

Keyword

nonn,easy

Author

Seiichi Manyama, Feb 07 2026

Status

approved