This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A004381 #50 Aug 16 2025 10:02:55
%S A004381 1,8,120,2024,35960,658008,12271512,231917400,4426165368,85113005120,
%T A004381 1646492110120,32006008361808,624668654531480,12233149001721760,
%U A004381 240260199935164200,4730523156632595024,93343021201262177400,1845382436487682488000
%N A004381 Binomial coefficient C(8n,n).
%D A004381 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 A004381 T. D. Noe, <a href="/A004381/b004381.txt">Table of n, a(n) for n=0..100</a>
%H A004381 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 A004381 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 A004381 From _Ilya Gutkovskiy_, Jan 16 2017: (Start)
%F A004381 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 A004381 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 A004381 a(n) ~ 2^(24*n+1)/(sqrt(Pi*n)*7^(7*n+1/2)). (End)
%F A004381 From _Peter Bala_, Feb 20 2022: (Start)
%F A004381 The o.g.f. A(x) is algebraic: (1 - A(x))*(1 + 7*A(x))^7 + (8^8)*x*A(x)^8 = 0.
%F A004381 Sum_{n >= 1} a(n)*( x*(7*x + 8)^7/(8^8*(1 + x)^8) )^n = x. (End)
%F A004381 From _Seiichi Manyama_, Aug 16 2025: (Start)
%F A004381 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(8*n+1,k).
%F A004381 G.f.: 1/(1 - 8*x*g^7) where g = 1+x*g^8 is the g.f. of A007556.
%F A004381 G.f.: g/(8-7*g) where g = 1+x*g^8 is the g.f. of A007556. (End)
%t A004381 Table[Binomial[8 n, n], {n, 0, 20}] (* _Vincenzo Librandi_, Aug 07 2014 *)
%o A004381 (Magma) [Binomial(8*n, n): n in [0..20]]; // _Vincenzo Librandi_, Aug 07 2014
%o A004381 (Python)
%o A004381 from math import comb
%o A004381 def A004381(n): return comb(n<<3,n) # _Chai Wah Wu_, Aug 01 2023
%Y A004381 Row 8 of A060539.
%Y A004381 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 A004381 nonn,easy
%O A004381 0,2
%A A004381 _N. J. A. Sloane_