A393276 a(n) = Sum_{k=0..floor(n/4)} binomial(2*k,k) * binomial(n-k-1,n-4*k).

1, 0, 0, 0, 2, 6, 12, 20, 36, 78, 182, 408, 866, 1802, 3804, 8208, 17874, 38802, 83778, 180660, 390636, 847490, 1841868, 4004124, 8704740, 18932398, 41213490, 89798064, 195789922, 427093530, 932052968, 2034966160, 4445125152, 9714318306, 21238410158, 46451025612, 101630144708

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)^3).

D-finite with recurrence: (2 + 4*n)*a(n) + (-11 - 3*n)*a(n + 1) + (-8 - 4*n)*a(n + 2) + (18 + 6*n)*a(n + 3) + (-4*n - 16)*a(n + 4) + (n + 5)*a(n + 5) = 0. - Robert Israel, Mar 08 2026

Maple

f:= gfun:-rectoproc({(2 + 4*n)*a(n) + (-11 - 3*n)*a(n + 1) + (-8 - 4*n)*a(n + 2) + (18 + 6*n)*a(n + 3) + (-4*n - 16)*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..35]); # Robert Israel, Mar 08 2026

Mathematica

Table[Sum[Binomial[2*k, k]*Binomial[n-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-k-1, n-4*k));
(Magma) [&+[Binomial(2*k, k)* Binomial(n-k-1, n-4*k) : k in [0..Floor(n/4)]] : n in [0..43] ]; // Vincenzo Librandi, Feb 11 2026

Crossrefs

Partial sums are A393246.

Cf. A360310, A393275, A393277.

Cf. A026585, A376809.

Sequence in context: A259470 A121315 A078878 * A095361 A095362 A288984

Adjacent sequences: A393273 A393274 A393275 * A393277 A393278 A393279

Keyword

nonn,easy

Author

Seiichi Manyama, Feb 08 2026

Status

approved