A030053 a(n) = binomial(2n+1,n-3).

1, 9, 55, 286, 1365, 6188, 27132, 116280, 490314, 2042975, 8436285, 34597290, 141120525, 573166440, 2319959400, 9364199760, 37711260990, 151584480450, 608359048206, 2438362177020, 9762479679106, 39049918716424, 156077261327400, 623404249591760

Offset: 3

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of UUUUUU's in all Dyck (n+3)-paths. - David Scambler, May 03 2013

Examples

			G.f. = x^3 + 9*x^4 + 55*x^5 + 286*x^6 + 1365*x^7 + 6188*x68 + ...
		

Links

• T. D. Noe, Table of n, a(n) for n = 3..200
• Milan Janjic, Two Enumerative Functions.
• Asamoah Nkwanta and Earl R. Barnes, Two Catalan-type Riordan Arrays and their Connections to the Chebyshev Polynomials of the First Kind, Journal of Integer Sequences, Vol. 15 (2012), Article 12.3.3. - From _N. J. A. Sloane_, Sep 16 2012.

Crossrefs

Diagonal 8 of triangle A100257.

Cf. A001622, A113187 (unsigned fourth column).

Cf. binomial(2*n+m, n): A000984 (m = 0), A001700 (m = 1), A001791 (m = 2), A002054 (m = 3), A002694 (m = 4), A003516 (m = 5), A002696 (m = 6), A030054 - A030056, A004310 - A004318.

Programs

Magma

[Binomial(2*n+1,n-3): n in [3..30]]; // Vincenzo Librandi, Aug 11 2015

Mathematica

Table[Binomial[2*n + 1, n - 3], {n, 3, 20}] (* T. D. Noe, Apr 03 2014 *)
Rest[Rest[Rest[CoefficientList[Series[128 x^3 / ((1 - Sqrt[1 - 4 x])^7 Sqrt[1 - 4 x]) + (-1 / x^4 + 5 / x^3 - 6 / x^2 + 1 / x), {x, 0, 40}], x]]]] (* Vincenzo Librandi, Aug 11 2015 *)

PARI

a(n) = binomial(2*n+1,n-3); \\ Joerg Arndt, May 08 2013

Formula

G.f.: x^3*128/((1-sqrt(1-4*x))^7*sqrt(1-4*x))+(-1/x^4+5/x^3-6/x^2+1/x). - Vladimir Kruchinin, Aug 11 2015

D-finite with recurrence: -(n+4)*(n-3)*a(n) +2*n*(2*n+1)*a(n-1)=0. - R. J. Mathar, Jan 28 2020

G.f.: x^3* 2F1(4,9/2;8;4x). - R. J. Mathar, Jan 28 2020

From Amiram Eldar, Jan 24 2022: (Start)

Sum_{n>=3} 1/a(n) = 22*Pi/(9*sqrt(3)) - 33/10.

Sum_{n>=3} (-1)^(n+1)/a(n) = 852*log(phi)/(5*sqrt(5)) - 1073/30, where phi is the golden ratio (A001622). (End)

From Peter Bala, Oct 13 2024: (Start)

a(n) = Integral_{x = 0..4} x^n * w(x) dx, where the weight function w(x) = (1/(2*Pi)) * sqrt(x)*(x^3 - 7*x^2 + 14*x - 7)/sqrt((4 - x)).

G.f. x^3 * B(x) * C(x)^7, where B(x) = 1/sqrt(1 - 4*x) is the g.f. of the central binomial coefficients A000984 and C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)