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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete Mathematics, Addison-Wesley, Reading, 2nd ed. 1994.

Links

• T. D. Noe, Table of n, a(n) for n = 0..100

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).

Programs

Magma

[ Binomial(5*n,n): n in [0..100] ]; // Vincenzo Librandi, Apr 13 2011

Maple

f := n->(5*n)!/((4*n)!*(n)!);

Mathematica

Table[ Binomial[5n, n], {n, 0, 18} ]

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

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)

From Peter Bala, Oct 17 2024: (Start)

Let G******(x) denote the o.g.f. of sequence A******.

For n >= 1 , a(n) = (5/2) * [x^n] G006013(x)^n.

For n >= 1, a(n) = [x^n] (1 + x)^(5*n) = (5/4) * [x^n] (1/(1 - x))^(4*n) = (5/3) * [x^n] G000108(x)^(3*n) = (5/2) * [x^n] G001764(x)^(2*n) = 5 * [x^n] G002293(x)^n.

a(n) = 5 * [x^n] (1 - G006632(x))^(-n) = (5/2) * [x^n] (1 - x*G006013(x))^(-2*n) = (5/3) * [x^n] (1 - x*G000108(x))^(-3*n) (apply Concrete Mathematics, equation 5.60, p. 201). (End)