A389285 G.f. A(x) satisfies A(x) = 1 + x/(1-x^3) * A(x)^3.

1, 1, 3, 12, 56, 279, 1464, 7973, 44637, 255315, 1485530, 8764851, 52317291, 315355879, 1916894598, 11736741483, 72318894814, 448109853543, 2790448493604, 17453874339997, 109608440551239, 690819127557024, 4368270982686774, 27704805819149325, 176194389879270093

Offset

0,3

Links

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

Formula

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

Mathematica

Table[Sum[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 *)
terms = 25; A[_] = 0; Do[A[x_] =1 + x /(1-x^3)*A[x]^3 + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Nov 16 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\3, binomial(n-2*k-1, k)*binomial(3*(n-3*k), n-3*k)/(2*(n-3*k)+1));
(Magma) [&+[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. A307678, A389196.

Cf. A001764, A376574.

Sequence in context: A379160 A390486 A386496 * A176281 A050147 A259800

Adjacent sequences: A389282 A389283 A389284 * A389286 A389287 A389288

Keyword

nonn,easy

Author

Seiichi Manyama, Oct 26 2025

Status

approved