OFFSET
3,2
COMMENTS
Number of UUUUUU's in all Dyck (n+3)-paths. - David Scambler, May 03 2013
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.
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)).
EXAMPLE
G.f. = x^3 + 9*x^4 + 55*x^5 + 286*x^6 + 1365*x^7 + 6188*x68 + ...
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 *)
PROG
(PARI) a(n) = binomial(2*n+1, n-3); \\ Joerg Arndt, May 08 2013
(Magma) [Binomial(2*n+1, n-3): n in [3..30]]; // Vincenzo Librandi, Aug 11 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved