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A008979 a(n) = (6n)!/(n!)^6.

%I A008979 #36 Aug 17 2024 09:01:21
%S A008979 1,720,7484400,137225088000,3246670537110000,88832646059788350720,
%T A008979 2670177736637149247308800,85722533226982363751829504000,
%U A008979 2889253496242619386328267523990000,101097362223624462291180422369532000000,3644153415887633116359073848179365185734400,134567406165969006655507763343147223231094784000
%N A008979 a(n) = (6n)!/(n!)^6.
%H A008979 Vincenzo Librandi, <a href="/A008979/b008979.txt">Table of n, a(n) for n = 0..100</a>
%H A008979 R. Mestrovic, <a href="http://arxiv.org/abs/1111.3057">Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011)</a>, arXiv:1111.3057 [math.NT], 2011.
%F A008979 From _Peter Bala_, Jul 12 2016: (Start)
%F A008979 a(n) = binomial(2*n,n)*binomial(3*n,n)*binomial(4*n,n)*
%F A008979   binomial(5*n,n)*binomial(6*n,n) = ( [x^n](1 + x)^(2*n) ) * ( [x^n](1 + x)^(3*n) ) * ( [x^n](1 + x)^(4*n) ) * ( [x^n](1 + x)^(5*n) ) * ( [x^n](1 + x)^(6*n) ) = [x^n](F(x)^(720*n)), where F(x) = 1 + x + 4478*x^2 + 53085611*x^3 + 926072057094*x^4 + 19977558181209910*x^5 + 493286693783478576177*x^6 + ... appears to have integer coefficients. For similar results see A000897, A002894, A002897, A006480, A008977, A008978, A186420 and A188662. (End)
%F A008979 a(n) ~ 3^(6*n+1/2)*4^(3*n-1)/(Pi*n)^(5/2). - _Ilya Gutkovskiy_, Jul 12 2016
%F A008979 From _Peter Bala_, Feb 14 2020: (Start)
%F A008979 a(m*p^k) == a(m*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers m and k - apply Mestrovic, equation 39, p. 12.
%F A008979 a(n) = [(x*y*z*u*v)^n] (1 + x + y + z + u + v)^(6*n). (End)
%p A008979 seq( (6*n)!/(n!)^6, n=0..20); # _G. C. Greubel_, Feb 17 2020
%t A008979 Table[(6 n)!/(n)!^6, {n, 0, 20}] (* _Vincenzo Librandi_, Aug 13 2014 *)
%o A008979 (Magma) [Factorial(6*n)/Factorial(n)^6: n in [0..20]]; // _Vincenzo Librandi_, Aug 13 2014
%o A008979 (PARI) vector(21, n, my(m=n-1); (6*m)!/(m!)^6 ) \\ _G. C. Greubel_, Feb 17 2020
%o A008979 (Sage) [factorial(6*n)/factorial(n)^6 for n in (0..20)] # _G. C. Greubel_, Feb 17 2020
%Y A008979 Cf. A000897, A000984, A002894, A002897, A006480, A008977, A008978, A071549, A071550, A071551, A071552, A186420, A188662.
%K A008979 nonn,easy
%O A008979 0,2
%A A008979 _N. J. A. Sloane_