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%I #49 Sep 08 2022 08:44:29
%S 1,5,45,455,4845,53130,593775,6724520,76904685,886163135,10272278170,
%T 119653565850,1399358844975,16421073515280,193253756909160,
%U 2280012686716080,26958221130508525
%N Binomial coefficients binomial(5n,n).
%H T. D. Noe, <a href="/A001449/b001449.txt">Table of n, a(n) for n = 0..100</a>
%F a(n) = (5*n)!/((4*n)!*(n)!).
%F a(n) is asymptotic to c*(3125/256)^n/sqrt(n), with c = sqrt(5/(8*Pi)) = 0.44603102903819277863474159... - _Benoit Cloitre_, Jan 23 2008
%F 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
%F G.f.: A(x) = x*B'(x)/B(x), where B(x)+1 is g.f. of A002294. - _Vladimir Kruchinin_, Oct 05 2015
%F From _Ilya Gutkovskiy_, Jan 16 2017: (Start)
%F O.g.f.: 4F3(1/5,2/5,3/5,4/5; 1/4,1/2,3/4; 3125*x/256).
%F E.g.f.: 4F4(1/5,2/5,3/5,4/5; 1/4,1/2,3/4,1; 3125*x/256). (End)
%F a(n) = hypergeom([-4*n, -n], [1], 1). - _Peter Luschny_, Mar 19 2018
%F From _Peter Bala_, Feb 20 2022: (Start)
%F 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).
%F The o.g.f. A(x) is algebraic: (1 - A(x))*(1 + 4*A(x))^4 + 3125*x*A(x)^5 = 0.
%F Sum_{n >= 1} a(n)*( x*(4*x + 5)^4/(3125*(1 + x)^5) )^n = x. (End)
%p f := n->(5*n)!/((4*n)!*(n)!);
%t Table[ Binomial[5n, n], {n, 0, 18} ]
%o (Magma)[ Binomial(5*n,n): n in [0..100] ]; // _Vincenzo Librandi_, Apr 13 2011
%o (Maxima)
%o B(x):=sum(binomial(5*n,n-1)/n*x^n,n,1,30);
%o taylor(x*diff(B(x),x)/B(x),x,0,10); /* _Vladimir Kruchinin_, Oct 05 2015 */
%o (PARI) a(n) = binomial(5*n, n) \\ _Altug Alkan_, Oct 05 2015
%Y 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).
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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