A051944 a(n) = C(n)*(4*n+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

Author

Barry E. Williams, Dec 20 1999

Keywords

• easy
• nonn
• changed

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

Crossrefs

Column k=4 of A330965.

Cf. A016777, A000108, A051924.

Programs

Magma

[Catalan(n)*(4*n+1):n in [0..30] ]; // Marius A. Burtea, Jan 05 2020

Magma

R:=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

Mathematica

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

PARI

{a(n)=if(n<0, 0, (4*n+1)*binomial(2*n,n)/(n+1))} /* Michael Somos, Sep 17 2006 */

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

From Peter Bala, Aug 23 2025: (Start)

a(n) = binomial(2*n, n) + 3*binomial(2*n, n-1) = A000984(n) + 3*A001791(n).

a(n) ~ 4^(n+1)/sqrt(Pi*n). (End)

Extensions

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