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A389873

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

1, 1, 2, 5, 16, 50, 162, 542, 1862, 6521, 23194, 83564, 304340, 1118643, 4144374, 15460134, 58021446, 218918296, 829928124, 3159728783, 12076125224, 46314516352, 178189745050, 687552897730, 2660012928960, 10316307315965, 40100222337016, 156199365141761

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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)} binomial(2*(n-3*k),k) * Catalan(n-3*k).

MATHEMATICA

Table[Sum[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, binomial(2*(n-3*k), k)*binomial(2*(n-3*k), n-3*k)/(n-3*k+1));