A387063 - OEIS

%I #19 Oct 30 2025 10:35:32

%S 1,1,2,7,22,75,264,958,3558,13463,51722,201223,791192,3139131,

%T 12552204,50533205,204654950,833219825,3408319290,14000834736,

%U 57732586382,238884001905,991558312956,4127602773465,17227553257504,72078391054381,302247212118142,1270059248710872,5347199223168084

%N G.f. A(x) satisfies A(x) = 1 + x/(1-x^2)^2 * A(x)^2.

%H Vincenzo Librandi, <a href="/A387063/b387063.txt">Table of n, a(n) for n = 0..1500</a>

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

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

%t Table[Sum[ Binomial[2*n-3*k-1,k]*CatalanNumber[n-2*k],{k,0,Floor[n/2]}],{n,0,30}] (* _Vincenzo Librandi_, Oct 30 2025 *)

%o (PARI) a(n) = sum(k=0, n\2, binomial(2*n-3*k-1, k)*binomial(2*(n-2*k), n-2*k)/(n-2*k+1));

%o (Magma) [&+[Catalan(n-2*k) * Binomial(2*n-3*k-1,k): k in [0..Floor(n/2)]] : n in [0..30] ]; // _Vincenzo Librandi_, Oct 30 2025

%Y Cf. A006319, A390103.

%Y Cf. A085139, A390102.

%Y Cf. A000108.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Oct 25 2025