A001449 Binomial coefficients binomial(5n,n).

1, 5, 45, 455, 4845, 53130, 593775, 6724520, 76904685, 886163135, 10272278170, 119653565850, 1399358844975, 16421073515280, 193253756909160, 2280012686716080, 26958221130508525

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

binomial(k*n,n): A000984 (k = 2), A005809 (k = 3), A005810 (k = 4), A004355 (k = 6), A004368 (k = 7), A004381 (k = 8), A169958 - A169961 (k = 9 thru 12).

Sequence in context: A093140 A137233 A357205 * A371756 A209442 A371519

Adjacent sequences: A001446 A001447 A001448 * A001450 A001451 A001452

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved