A393247 a(n) = Sum_{k=0..floor(n/5)} binomial(2*k,k) * binomial(n-2*k,3*k).

1, 1, 1, 1, 1, 3, 9, 21, 41, 71, 119, 211, 409, 835, 1701, 3365, 6473, 12307, 23539, 45691, 89791, 177205, 348835, 683503, 1335441, 2610099, 5114439, 10050315, 19784709, 38964103, 76711165, 150980413, 297203129, 585404659, 1153999639, 2276421527, 4492549499

Offset

0,6

Links

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

Formula

G.f.: 1/sqrt((1-x)^2 - 4*x^5/(1-x)).

D-finite with recurrence: (8 + 4*n)*a(n) + (-14 - 4*n)*a(n + 1) + (n + 3)*a(n + 2) + (-15 - 4*n)*a(n + 3) + (27 + 6*n)*a(n + 4) + (-21 - 4*n)*a(n + 5) + (n + 6)*a(n + 6) = 0. - Robert Israel, Feb 09 2026

Maple

f:= gfun:-rectoproc({(8 + 4*n)*a(n) + (-14 - 4*n)*a(n + 1) + (n + 3)*a(n + 2) + (-15 - 4*n)*a(n + 3) + (27 + 6*n)*a(n + 4) + (-21 - 4*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-2*k, 3*k], {k, 0, Floor[n/5]}], {n, 0, 35}] (* Vincenzo Librandi, Feb 08 2026 *)

Prog

(PARI) a(n) = sum(k=0, n\5, binomial(2*k, k)*binomial(n-2*k, 3*k));
(Magma) [&+[Binomial(2*k, k)* Binomial(n-2*k, 3*k) : k in [0..Floor(n/5)]] : n in [0..35] ]; // Vincenzo Librandi, Feb 08 2026

Crossrefs

Cf. A387599, A393249.

Sequence in context: A064999 A100135 A393250 * A024173 A396151 A097119

Adjacent sequences: A393244 A393245 A393246 * A393248 A393249 A393250

Keyword

nonn,easy

Author

Seiichi Manyama, Feb 07 2026

Status

approved