A393251 a(n) = Sum_{k=0..floor(n/6)} binomial(2*k,k) * binomial(n-4*k,2*k).

1, 1, 1, 1, 1, 1, 3, 7, 13, 21, 31, 43, 63, 103, 181, 321, 553, 913, 1463, 2331, 3771, 6243, 10513, 17773, 29851, 49651, 82033, 135413, 224433, 374121, 626283, 1049695, 1757245, 2935765, 4898143, 8172667, 13652603, 22842771, 38265153, 64131837, 107474983

Offset

0,7

Links

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

Formula

a(n) = Sum_{k=0..floor(n/6)} multinomial(n-4*k; k, k, n-6*k).

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

D-finite with recurrence: (12 + 4*n)*a(n) + (-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) + (-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) = 1}, a(n), remember):
map(f, [$0..40]); # Robert Israel, Feb 09 2026

Mathematica

Table[Sum[Binomial[2*k, k]*Binomial[n-4*k, 2*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-4*k, 2*k));
(Magma) [&+[Binomial(2*k, k)* Binomial(n-4*k, 2*k) : k in [0..Floor(n/6)]] : n in [0..43] ]; // Vincenzo Librandi, Feb 10 2026

Crossrefs

Cf. A002426, A098479, A098482, A387599.

Cf. A393250.

Sequence in context: A247890 A353887 A063541 * A206246 A171965 A011898

Adjacent sequences: A393248 A393249 A393250 * A393252 A393253 A393254

Keyword

nonn,easy

Author

Seiichi Manyama, Feb 07 2026

Status

approved