A051945 a(n) = C(n)*(5*n+1) where C(n) = Catalan numbers (A000108).

1, 6, 22, 80, 294, 1092, 4092, 15444, 58630, 223652, 856596, 3292016, 12688732, 49031400, 189885240, 736808220, 2863971270, 11149451940, 43465121700, 169657266240, 662976162420, 2593424304120, 10154564564040, 39794915183400, 156078401826204, 612605246582952

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=5 of A330965.

Cf. A016813, A000108, A051924.

Programs

Magma

[Catalan(n)*(5*n+1):n in [0..27] ]; // Marius A. Burtea, Jan 05 2020

Magma

R:=PowerSeriesRing(Rationals(),29); (Coefficients(R!((2-3*x-2*Sqrt(1-4*x))/(x*Sqrt(1-4*x))))); // Marius A. Burtea, Jan 05 2020

Mathematica

Table[CatalanNumber[n](5n+1),{n,0,30}] (* Harvey P. Dale, Jul 27 2020 *)

PARI

a(n) = (5*n+1)*binomial(2*n, n)/(n+1)  \\ Michel Marcus, Jul 12 2013

Formula

(n+1)*(5n-4)*a(n) - 2*(5n+1)(2n-1)*a(n-1) = 0. - R. J. Mathar, Jul 09 2012

G.f.: (2 - 3*x - 2*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) + 4*binomial(2*n, n-1) = A000984(n) + 4*A001791(n).

a(n) ~ 4^n * 5/sqrt(Pi*n). (End)

E.g.f.: exp(2*x)*((1 + 5*x)*BesselI(0, 2*x) - BesselI(1, 2*x) - 5*x*BesselI(2, 2*x)). - Stefano Spezia, Aug 29 2025

Extensions

Terms a(21) and beyond from Andrew Howroyd, Jan 04 2020