A383573 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k) * binomial(2*(n-2*k),n-2*k).

1, 2, 7, 24, 89, 338, 1311, 5152, 20449, 81778, 328999, 1330008, 5398265, 21984610, 89791103, 367643776, 1508560257, 6201927074, 25540266503, 105336838616, 435035342553, 1798875915826, 7446653956895, 30857577536800, 127987031688161, 531301328367762, 2207281722474919

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Formula

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

a(n) ~ phi^(3*n + 3) / (2^(3/2) * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, May 01 2025

Mathematica

Table[Sum[Binomial[n-k, k]* Binomial[2*(n-2*k), n-2*k], {k, 0, Floor[n/2]}], {n, 0, 30}] (* Vincenzo Librandi, May 03 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\2, binomial(n-k, k)*binomial(2*(n-2*k), n-2*k));
(Magma) [&+[Binomial(n-k, k) * Binomial(2*(n-2*k), n-2*k): k in [0..Floor(n div 2)]]: n in [0..35]]; // Vincenzo Librandi, May 03 2025

Crossrefs

Cf. A026375, A383581, A383582.

Cf. A360290.

Sequence in context: A151293 A122446 A150390 * A052705 A150391 A378585

Adjacent sequences: A383570 A383571 A383572 * A383574 A383575 A383576

Keyword

nonn

Author

Seiichi Manyama, Apr 30 2025

Status

approved