OFFSET
0,2
LINKS
T. D. Noe, Table of n, a(n) for n = 0..100
FORMULA
a(n) = (5*n)!/((4*n)!*(n)!).
a(n) is asymptotic to c*(3125/256)^n/sqrt(n), with c = sqrt(5/(8*Pi)) = 0.44603102903819277863474159... - Benoit Cloitre, Jan 23 2008
a(n) = C(5*n-1,n-1)*C(25*n^2,2)/(3*n*C(5*n+1,3)), n>0. - Gary Detlefs, Jan 02 2014
G.f.: A(x) = x*B'(x)/B(x), where B(x)+1 is g.f. of A002294. - Vladimir Kruchinin, Oct 05 2015
From Ilya Gutkovskiy, Jan 16 2017: (Start)
O.g.f.: 4F3(1/5,2/5,3/5,4/5; 1/4,1/2,3/4; 3125*x/256).
E.g.f.: 4F4(1/5,2/5,3/5,4/5; 1/4,1/2,3/4,1; 3125*x/256). (End)
a(n) = hypergeom([-4*n, -n], [1], 1). - Peter Luschny, Mar 19 2018
From Peter Bala, Feb 20 2022: (Start)
4*n(4*n-1)*(4*n-2)*(4*n-3)*a(n) = 5*(5*n-1)*(5*n-2)*(5*n-3)*(5*n-4)*a(n-1).
The o.g.f. A(x) is algebraic: (1 - A(x))*(1 + 4*A(x))^4 + 3125*x*A(x)^5 = 0.
Sum_{n >= 1} a(n)*( x*(4*x + 5)^4/(3125*(1 + x)^5) )^n = x. (End)
MAPLE
f := n->(5*n)!/((4*n)!*(n)!);
MATHEMATICA
Table[ Binomial[5n, n], {n, 0, 18} ]
PROG
(Magma)[ Binomial(5*n, n): n in [0..100] ]; // Vincenzo Librandi, Apr 13 2011
(Maxima)
B(x):=sum(binomial(5*n, n-1)/n*x^n, n, 1, 30);
taylor(x*diff(B(x), x)/B(x), x, 0, 10); /* Vladimir Kruchinin, Oct 05 2015 */
(PARI) a(n) = binomial(5*n, n) \\ Altug Alkan, Oct 05 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved