A389890 G.f. A(x) satisfies A(x) = 1 + x*(1+x^3)^2*A(x)^3.

1, 1, 3, 12, 57, 285, 1500, 8193, 46011, 263991, 1540783, 9119145, 54601746, 330153384, 2013112923, 12364421871, 76425011259, 475036074345, 2967394228125, 18618851162820, 117291191536701, 741560529250605, 4703846060352603, 29926800398213649, 190923429145757460

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(2*(n-3*k),k) * A001764(n-3*k).

Mathematica

Table[Sum[Binomial[2*(n-3*k), 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 15 2025 *)
terms = 25; A[_] = 0; Do[A[x_] =1 + x *(1+x^3)^2*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(2*(n-3*k), k)*binomial(3*(n-3*k), n-3*k)/(2*(n-3*k)+1));
(Magma) [&+[Binomial(2*(n-3*k), 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 15 2025

Crossrefs

Cf. A366221, A385065.

Cf. A001764.

Sequence in context: A110309 A263667 A101106 * A389286 A165310 A133158

Adjacent sequences: A389887 A389888 A389889 * A389891 A389892 A389893

Keyword

nonn,easy

Author

Seiichi Manyama, Oct 18 2025

Status

approved