A050479 a(n) = C(n)*(9*n + 1) where C(n) = Catalan numbers (A000108).

1, 10, 38, 140, 518, 1932, 7260, 27456, 104390, 398684, 1528436, 5878600, 22673308, 87662200, 339653880, 1318498920, 5126862150, 19965297660, 77855108100, 303969268680, 1188105796020, 4648590733800, 18205030164360, 71356399639200, 279909199969308, 1098799886728152

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=9 of A330965.

Cf. A017077, A000108, A051945.

Programs

Magma

[Catalan(n)*(9*n+1):n in [0..27] ]; // Marius A. Burtea, Jan 05 2020

Magma

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

Mathematica

A050479[n_] := CatalanNumber[n]*(9*n + 1);
Array[A050479, 30, 0] (* Paolo Xausa, Aug 24 2025 *)

Formula

4*(n+1)*a(n) + (-23*n-1)*a(n-1) + 14*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Feb 13 2015

-(n+1)*(9*n-8)*a(n) + 2*(9*n+1)*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Feb 13 2015

G.f.: (4 - 7*x - 4*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) + 8*binomial(2*n, n-1) = A000984(n) + 8*A001791(n).

a(n) ~ 4^n * 9/sqrt(Pi*n). (End)

Extensions

Terms a(21) and beyond from Andrew Howroyd, Jan 05 2020