A051944 a(n) = C(n)*(4n+1) where C(n) = Catalan numbers (A000108).

1, 5, 18, 65, 238, 882, 3300, 12441, 47190, 179894, 688636, 2645370, 10192588, 39373700, 152443080, 591385545, 2298248550, 8945490510, 34867625100, 136079265630, 531693754020, 2079632696700, 8141948163960, 31904544069450, 125120702290428, 491056586546652

Offset

0,2

References

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

Links

Andrew Howroyd, Table of n, a(n) for n = 0..200

Formula

The Hankel determinant transform is A025172(n-1). - Michael Somos, Sep 17 2006

-(n+1)*(4*n-3)*a(n) + 2*(4*n+1)*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Nov 19 2014

G.f.: (3 - 4*x - 3*sqrt(1 - 4*x))/(2*x*sqrt(1 - 4*x)). - Ilya Gutkovskiy, Jun 13 2017

Mathematica

Table[CatalanNumber[n](4n+1), {n, 0, 30}] (* Harvey P. Dale, Feb 21 2022 *)

Prog

(PARI) {a(n)=if(n<0, 0, (4*n+1)*binomial(2*n, n)/(n+1))} /* Michael Somos, Sep 17 2006 */
(Magma) [Catalan(n)*(4*n+1):n in [0..30] ]; // Marius A. Burtea, Jan 05 2020
(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

Crossrefs

Column k=4 of A330965.

Cf. A016777, A000108, A051924.

Sequence in context: A222373 A147535 A184309 * A153373 A225015 A371871

Adjacent sequences: A051941 A051942 A051943 * A051945 A051946 A051947

Keyword

easy,nonn

Author

Barry E. Williams, Dec 20 1999

Extensions

Terms a(21) and beyond from Andrew Howroyd, Jan 02 2020

Status

approved