%I #39 Aug 01 2023 09:55:42

%S 1,8,120,2024,35960,658008,12271512,231917400,4426165368,85113005120,

%T 1646492110120,32006008361808,624668654531480,12233149001721760,

%U 240260199935164200,4730523156632595024

%N Binomial coefficient C(8n,n).

%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

%H T. D. Noe, <a href="/A004381/b004381.txt">Table of n, a(n) for n=0..100</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%F a(n) = C(8*n-1,n-1)*C(64*n^2,2)/(3*n*C(8*n+1,3)), n>0. - _Gary Detlefs_, Jan 02 2014

%F From _Ilya Gutkovskiy_, Jan 16 2017: (Start)

%F O.g.f.: 7F6(1/8,1/4,3/8,1/2,5/8,3/4,7/8; 1/7,2/7,3/7,4/7,5/7,6/7; 16777216*x/823543).

%F E.g.f.: 7F7(1/8,1/4,3/8,1/2,5/8,3/4,7/8; 1/7,2/7,3/7,4/7,5/7,6/7,1; 16777216*x/823543).

%F a(n) ~ 2^(24*n+1)/(sqrt(Pi*n)*7^(7*n+1/2)). (End)

%F From _Peter Bala_, Feb 20 2022: (Start)

%F The o.g.f. A(x) is algebraic: (1 - A(x))*(1 + 7*A(x))^7 + (8^8)*x*A(x)^8 = 0.

%F Sum_{n >= 1} a(n)*( x*(7*x + 8)^7/(8^8*(1 + x)^8) )^n = x. (End)

%t Table[Binomial[8 n, n], {n, 0, 20}] (* _Vincenzo Librandi_, Aug 07 2014 *)

%o (Magma) [Binomial(8*n, n): n in [0..20]]; // _Vincenzo Librandi_, Aug 07 2014

%o (Python)

%o from math import comb

%o def A004381(n): return comb(n<<3,n) # _Chai Wah Wu_, Aug 01 2023

%Y binomial(k*n,n): A000984 (k = 2), A005809 (k = 3), A005810 (k = 4), A001449 (k = 5), A004355 (k = 6), A004368 (k = 7), A169958 - A169961 (k = 9 thru 12).

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_