A050477 a(n) = C(n)*(7*n + 1) where C(n) = Catalan numbers (A000108).

1, 8, 30, 110, 406, 1512, 5676, 21450, 81510, 311168, 1192516, 4585308, 17681020, 68346800, 264769560, 1027653570, 3995416710, 15557374800, 60660114900, 236813267460, 925540979220, 3621007518960, 14179797364200, 55575657411300, 217993800897756, 855702566655552

Offset: 0

Author

Barry E. Williams, Dec 24 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=7 of A330965.

Cf. A016921, A000108, A051945.

Programs

Magma

[Catalan(n)*(7*n+1):n in [0..25] ]; // Marius A. Burtea, Jan 05 2020

Magma

R:=PowerSeriesRing(Rationals(),27); (Coefficients(R!( (3-5*x-3*Sqrt(1-4*x))/(x*Sqrt(1 - 4*x))) )); // Marius A. Burtea, Jan 05 2020

Mathematica

Table[CatalanNumber[n](7n+1),{n,0,30}] (* Harvey P. Dale, Jun 01 2024 *)

PARI

a(n) = (7*n+1) * binomial(2*n,n)/(n+1) \\ Michel Marcus, Jul 24 2013

Formula

3*(n+1)*a(n) + (-17*n-1)*a(n-1) + 10*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Feb 13 2015

-(n+1)*(7*n-6)*a(n) + 2*(7*n+1)*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Feb 13 2015

G.f.: (3 - 5*x - 3*sqrt(1 - 4*x))/(x*sqrt(1 - 4*x)). - Ilya Gutkovskiy, Jun 13 2017

From Peter Bala, Aug 23 2025: (Start)

a(n) = binomial(2*n, n) + 6*binomial(2*n, n-1) = A000984(n) + 6*A001791(n).

a(n) ~ 4^n * 7/sqrt(Pi*n). (End)

Extensions

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