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a(n) = C(n)*(4n+1) where C(n) = Catalan numbers (A000108).
4

%I #27 Sep 08 2022 08:44:59

%S 1,5,18,65,238,882,3300,12441,47190,179894,688636,2645370,10192588,

%T 39373700,152443080,591385545,2298248550,8945490510,34867625100,

%U 136079265630,531693754020,2079632696700,8141948163960,31904544069450,125120702290428,491056586546652

%N a(n) = C(n)*(4n+1) where C(n) = Catalan numbers (A000108).

%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

%H Andrew Howroyd, <a href="/A051944/b051944.txt">Table of n, a(n) for n = 0..200</a>

%F The Hankel determinant transform is A025172(n-1). - _Michael Somos_, Sep 17 2006

%F -(n+1)*(4*n-3)*a(n) + 2*(4*n+1)*(2*n-1)*a(n-1) = 0. - _R. J. Mathar_, Nov 19 2014

%F G.f.: (3 - 4*x - 3*sqrt(1 - 4*x))/(2*x*sqrt(1 - 4*x)). - _Ilya Gutkovskiy_, Jun 13 2017

%t Table[CatalanNumber[n](4n+1),{n,0,30}] (* _Harvey P. Dale_, Feb 21 2022 *)

%o (PARI) {a(n)=if(n<0, 0, (4*n+1)*binomial(2*n,n)/(n+1))} /* _Michael Somos_, Sep 17 2006 */

%o (Magma) [Catalan(n)*(4*n+1):n in [0..30] ]; // _Marius A. Burtea_, Jan 05 2020

%o (Magma) R<x>:=PowerSeriesRing(Rationals(),30); (Coefficients(R!( (3 - 4*x - 3*Sqrt(1 - 4*x))/(2*x*Sqrt(1 - 4*x)))) ); // _Marius A. Burtea_, Jan 05 2020

%Y Column k=4 of A330965.

%Y Cf. A016777, A000108, A051924.

%K easy,nonn

%O 0,2

%A _Barry E. Williams_, Dec 20 1999

%E Terms a(21) and beyond from _Andrew Howroyd_, Jan 02 2020