A390171 G.f. A(x) satisfies A(x) = 1 + x/(1+x^3) * A(x)^3.

1, 1, 3, 12, 54, 267, 1392, 7533, 41907, 238179, 1377000, 8072619, 47876901, 286739379, 1731753408, 10534970955, 64495997604, 397062007827, 2456624420430, 15266724660549, 95254588766973, 596475960446748, 3747346172672004, 23613159856210485, 149202136625479449, 945128700216537903

Offset

0,3

Links

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

Formula

a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(n-2*k-1,k) * A001764(n-3*k).

Mathematica

Table[Sum[(-1)^k*Binomial[n-2*k-1, k]*Binomial[3*(n-3*k), n-3*k]/(2*(n-3*k)+1), {k, 0, Floor[n/3]}], {n, 0, 25}] (* Vincenzo Librandi, Nov 14 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\3, (-1)^k*binomial(n-2*k-1, k)*binomial(3*(n-3*k), n-3*k)/(2*(n-3*k)+1));
(Magma) [&+[(-1)^k*Binomial(n-2*k-1, k)*Binomial(3*(n-3*k), n-3*k)/(2*(n-3*k)+1): k in [0..Floor(n/3)]] : n in [0..30] ]; // Vincenzo Librandi, Nov 14 2025

Crossrefs

Cf. A386496, A389890, A390172.

Cf. A001764, A389285.

Sequence in context: A193115 A270489 A388857 * A390518 A335819 A263853

Adjacent sequences: A390168 A390169 A390170 * A390172 A390173 A390174

Keyword

nonn

Author

Seiichi Manyama, Oct 28 2025

Status

approved