A393277 a(n) = Sum_{k=0..floor(n/4)} binomial(2*k,k) * binomial(n-2*k-1,n-4*k).

1, 0, 0, 0, 2, 4, 6, 8, 16, 36, 74, 136, 248, 476, 946, 1864, 3606, 6948, 13506, 26456, 51856, 101396, 198158, 387976, 761170, 1494636, 2935278, 5766264, 11335628, 22301596, 43901474, 86457032, 170330356, 335718772, 661994394, 1305896760, 2577024428, 5087143484, 10045529338

Offset

0,5

Links

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

Formula

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

D-finite with recurrence: (4 + 4*n)*a(n) + (-4*n - 12)*a(n + 1) + (-2 - n)*a(n + 2) + (9 + 3*n)*a(n + 3) + (-3*n - 12)*a(n + 4) + (n + 5)*a(n + 5) = 0. - Robert Israel, Mar 08 2026

Maple

f:= gfun:-rectoproc({(4 + 4*n)*a(n) + (-4*n - 12)*a(n + 1) + (-2 - n)*a(n + 2) + (9 + 3*n)*a(n + 3) + (-3*n - 12)*a(n + 4) + (n + 5)*a(n + 5), a(0) = 1, a(1) = 0, a(2) = 0, a(3) = 0, a(4) = 2}, a(n), remember):
map(f, [$0..40]); # Robert Israel, Mar 08 2026

Mathematica

Table[Sum[Binomial[2*k, k]*Binomial[n-2*k-1, n-4*k], {k, 0, Floor[n/4]}], {n, 0, 35}] (* Vincenzo Librandi, Feb 11 2026 *)

Prog

(PARI) a(n) = sum(k=0, n\4, binomial(2*k, k)*binomial(n-2*k-1, n-4*k));
(Magma) [&+[Binomial(2*k, k)* Binomial(n-2*k-1, n-4*k) : k in [0..Floor(n/4)]] : n in [0..43] ]; // Vincenzo Librandi, Feb 11 2026

Crossrefs

Partial sums are A098482.

Cf. A360310, A393275, A393276.

Sequence in context: A122408 A326037 A100055 * A334031 A304660 A344902

Adjacent sequences: A393274 A393275 A393276 * A393278 A393279 A393280

Keyword

nonn,easy

Author

Seiichi Manyama, Feb 08 2026

Status

approved