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

0,2

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

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)

Mathematica

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

Prog

(PARI) a(n) = (7*n+1) * binomial(2*n, n)/(n+1) \\ Michel Marcus, Jul 24 2013
(Magma) [Catalan(n)*(7*n+1):n in [0..25] ]; // Marius A. Burtea, Jan 05 2020
(Magma) R<x>:=PowerSeriesRing(Rationals(), 27); (Coefficients(R!( (3-5*x-3*Sqrt(1-4*x))/(x*Sqrt(1 - 4*x))) )); // Marius A. Burtea, Jan 05 2020

Crossrefs

Column k=7 of A330965.

Cf. A016921, A000108, A051945.

Sequence in context: A372252 A163613 A279217 * A239612 A055737 A293965

Adjacent sequences: A050474 A050475 A050476 * A050478 A050479 A050480

Keyword

easy,nonn

Author

Barry E. Williams, Dec 24 1999

Extensions

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

Status

approved