A390132 G.f. A(x) satisfies A(x) = 1 + x/(1+x^2)^2 * A(x)^2.

1, 1, 2, 3, 6, 15, 40, 110, 310, 891, 2602, 7699, 23032, 69547, 211692, 648861, 2000998, 6204133, 19328474, 60475420, 189950398, 598718725, 1893169916, 6003728385, 19090291408, 60851602621, 194409662846, 622411297884, 1996576883508, 6416310703231, 20654901049256

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1500

Formula

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

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

Mathematica

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

Prog

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

Crossrefs

Cf. A187256, A390133.

Cf. A000108, A387063.

Sequence in context: A147773 A006403 A129960 * A115098 A036418 A120589

Adjacent sequences: A390129 A390130 A390131 * A390133 A390134 A390135

Keyword

nonn

Author

Seiichi Manyama, Oct 26 2025

Status

approved