A050489 a(n) = C(n)*(10*n + 1) where C(n) = Catalan numbers (A000108).

1, 11, 42, 155, 574, 2142, 8052, 30459, 115830, 442442, 1696396, 6525246, 25169452, 97319900, 377096040, 1463921595, 5692584870, 22169259090, 86452604700, 337547269290, 1319388204420, 5162382341220, 20217646564440, 79246770753150, 310866899505084

Offset: 0

Author

Barry E. Williams, Dec 27 1999

Keywords

• easy
• nonn
• changed

References

• Albert 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=10 of A330965.

Cf. A017173, A027810, A000108.

Programs

Magma

[Catalan(n)*(10*n+1):n in [0..30] ]; // Marius A. Burtea, Jan 05 2020

Mathematica

Table[CatalanNumber[n](10n+1),{n,0,30}] (* Harvey P. Dale, Jul 19 2011 *)

PARI

a(n)=binomial(2*n,n)/(n+1)*(10*n+1) \\ Charles R Greathouse IV, Oct 23 2023

Formula

-(n+1)*(10*n-9)*a(n) + 2*(10*n+1)*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Dec 03 2014

From Stefano Spezia, Feb 16 2020: (Start)

O.g.f.: 2*(1 + sqrt(1 - 4*x) + 16*x)/((1 + sqrt(1 - 4*x))^2*sqrt(1 - 4*x)).

E.g.f.: exp(2*x)*(I_0(2*x) + 9*I_1(2*x)), where I_n(x) is the modified Bessel function of the first kind.

(End)

G.f.: (9 - 16*x - 9*sqrt(1 - 4*x))/(2*x*sqrt(1 - 4*x)). - Amiram Eldar, Jul 08 2023

From Peter Bala, Aug 23 2025: (Start)

a(n) = binomial(2*n, n) + 9*binomial(2*n, n-1) = A000984(n) + 9*A001791(n).

a(n) ~ 4^n * 10/sqrt(Pi*n). (End)

Extensions

Corrected and extended by Harvey P. Dale, Jul 19 2011