A389894 G.f. A(x) satisfies A(x) = 1 + x*(1-x^3)^2*A(x)^2.

1, 1, 2, 5, 12, 34, 102, 318, 1022, 3353, 11182, 37788, 129108, 445235, 1547710, 5417478, 19078334, 67548056, 240302580, 858541455, 3079197456, 11082288288, 40012944894, 144887840770, 526041042648, 1914572818653, 6984052883368, 25530203469473, 93507748002206

Offset

0,3

Links

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

Formula

G.f.: 2/(1 + sqrt(1 - 4*x*(1-x^3)^2)).

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

Mathematica

Table[Sum[(-1)^k* Binomial[2*(n-3*k), k]*CatalanNumber[n-3*k], {k, 0, Floor[n/3]}], {n, 0, 30}] (* Vincenzo Librandi, Oct 29 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\3, (-1)^k*binomial(2*(n-3*k), k)*binomial(2*(n-3*k), n-3*k)/(n-3*k+1));
(Magma) [&+[Catalan(n-3*k) * (-1)^k* Binomial(2*(n-3*k), k): k in [0..Floor(n/3)]] : n in [0..30] ]; // Vincenzo Librandi, Oct 29 2025

Crossrefs

Cf. A360271, A389895.

Cf. A000108.

Sequence in context: A032292 A151408 A121956 * A390134 A348102 A176638

Adjacent sequences: A389891 A389892 A389893 * A389895 A389896 A389897

Keyword

nonn,easy

Author

Seiichi Manyama, Oct 19 2025

Status

approved