A050478 a(n) = C(n)*(8*n+1) where C(n) = Catalan numbers (A000108).

1, 9, 34, 125, 462, 1722, 6468, 24453, 92950, 354926, 1360476, 5231954, 20177164, 78004500, 302211720, 1173076245, 4561139430, 17761336230, 69257611500, 270391268070, 1056823387620

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

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Crossrefs

Column k=8 of A330965.

Cf. A016993, A000108, A051945.

Programs

Magma

[Catalan(n)*(8*n+1):n in [0..30]]; // Vincenzo Librandi, Jan 27 2013

Magma

R:=PowerSeriesRing(Rationals(),30); (Coefficients(R!( (7-12*x-7*Sqrt(1-4*x))/(2*x*Sqrt(1-4*x))))); // Marius A. Burtea, Jan 05 2020

Mathematica

Table[CatalanNumber[n](8n+1),{n,0,20}] (* Harvey P. Dale, May 20 2012 *)

Formula

-(n+1)*(8*n-7)*a(n) + 2*(8*n+1)*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Dec 03 2014

G.f.: (7 - 12*x - 7*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) + 7*binomial(2*n, n-1) = A000984(n) + 7*A001791(n).

a(n) ~ 2^(2*n+3)/sqrt(Pi*n). (End)