A389945 G.f. A(x) satisfies A(x) = 1 + x*(1-x^3)^3*A(x)^3.

1, 1, 3, 12, 52, 255, 1320, 7095, 39213, 221403, 1271552, 7404933, 43624359, 259524942, 1556899818, 9407758965, 57208440741, 349830089730, 2149842159618, 13270295155380, 82240502676312, 511512720782229, 3191902869538211, 19977503222507118, 125378314629713370

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

Mathematica

Table[Sum[(-1)^k*Binomial[3*(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 13 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\3, (-1)^k*binomial(3*(n-3*k), k)*binomial(3*(n-3*k), n-3*k)/(2*(n-3*k)+1));
(Magma) [&+[(-1)^k*Binomial(3*(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 13 2025

Crossrefs

Cf. A001764, A389891.

Sequence in context: A000256 A274396 A299113 * A124202 A138269 A228771

Adjacent sequences: A389942 A389943 A389944 * A389946 A389947 A389948

Keyword

nonn,easy

Author

Seiichi Manyama, Oct 20 2025

Status

approved