OFFSET
4,2
LINKS
Robert Israel, Table of n, a(n) for n = 4..1661
Milan Janjic, Two Enumerative Functions.
FORMULA
G.f.: x^4*512/((1-sqrt(1-4*x))^9*sqrt(1-4*x))+(-1/x^5+7/x^4-15/x^3+10/x^2-1/x). - Vladimir Kruchinin, Aug 11 2015
From Robert Israel, Jun 11 2019: (Start)
(54 + 36*n)*a(n) + (-438 - 129*n)*a(n + 1) + (714 + 138*n)*a(n + 2) + (-432 - 63*n)*a(n + 3) + (110 + 13*n)*a(n + 4) + (-10 - n)*a(n + 5) = 0.
a(n) ~ 2^(2*n+1)/sqrt(n*Pi). (End)
From Amiram Eldar, Jan 24 2022: (Start)
Sum_{n>=4} 1/a(n) = 317/210 - 2*Pi/(9*sqrt(3)).
Sum_{n>=4} (-1)^n/a(n) = 2908*log(phi)/(5*sqrt(5)) - 8697/70, where phi is the golden ratio (A001622). (End)
G.f.: 2F1([11/2,5],[10],4*x). - Karol A. Penson, Apr 24 2024
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^4 - 9*x^3 + 27*x^2 - 30*x + 9)/sqrt((4 - x)).
G.f. x^4 * B(x) * C(x)^9, 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)
D-finite with recurrence -(n+5)*(n-4)*a(n) +2*n*(2*n+1)*a(n-1)=0. - R. J. Mathar, Nov 22 2024
MAPLE
seq(binomial(2*n+1, n-4), n=4..50); # Robert Israel, Jun 11 2019
MATHEMATICA
Table[Binomial[2n+1, n-4], {n, 4, 40}] (* Harvey P. Dale, Mar 31 2011 *)
PROG
(PARI) vector(30, n, m=n+4; binomial(2*m+1, m-4)) \\ Michel Marcus, Aug 11 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved