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%I #50 Jul 21 2023 17:26:50
%S 1,120,113400,168168000,305540235000,623360743125120,
%T 1370874167589326400,3177459078523411968000,7656714453153197981835000,
%U 19010638202652030712978200000,48334775757901219912115629238400,125285878026462826569986857692288000
%N a(n) = (5*n)!/(n!)^5.
%C Number of paths of length 5n in Z^5 from (0,0,0,0,0) to (n,n,n,n,n).
%H Vincenzo Librandi, <a href="/A008978/b008978.txt">Table of n, a(n) for n = 0..100</a>
%H V. Batyrev, <a href="http://www.ams.org/journal-getitem?pii=S0273-0979-00-00875-2">Review of "Mirror Symmetry and Algebraic Geometry"</a>, by D. A. Cox and S. Katz, Bull. Amer. Math. Soc., 37 (No. 4, 2000), 473-476.
%H R. M. Dickau, <a href="http://mathforum.org/advanced/robertd/path5d.html">5-D shortest path diagrams</a>
%H 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 a(n) ~ 5^(5*n+1/2) / (4 * Pi^2 * n^2). - _Vaclav Kotesovec_, Mar 07 2014
%F From _Peter Bala_, Jul 12 2016: (Start)
%F a(n) = binomial(2*n,n)*binomial(3*n,n)*binomial(4*n,n)*binomial(5*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]( F(x)^(120*n) ), where F(x) = 1 + x + 353*x^2 + 318986*x^3 + 408941594*x^4 + 633438203535*x^5 + 1105336091531052*x^6 + ... appears to have integer coefficients. For similar results see A000897, A002894, A002897, A006480, A008977, A186420 and A188662. (End)
%F From _Peter Bala_, Jul 17 2016: (Start)
%F a(n) = Sum_{k = 0..4*n} (-1)^k*binomial(5*n,n + k)*binomial(n + k,k)^5.
%F a(n) = Sum_{k = 0..5*n} (-1)^(n+k)*binomial(5*n,k)*binomial(n + k,k)^5. (End)
%F From _Ilya Gutkovskiy_, Nov 23 2017: (Start)
%F O.g.f.: 4F3(1/5,2/5,3/5,4/5; 1,1,1; 3125*x).
%F E.g.f.: 4F4(1/5,2/5,3/5,4/5; 1,1,1,1; 3125*x). (End)
%F From _Peter Bala_, Feb 16 2020: (Start)
%F 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 a(n) = [(x*y*z*u)^n] (1 + x + y + z + u )^(5*n). (End)
%F a(n) = 120*A322252(n). - _R. J. Mathar_, Jun 21 2023
%F a(n) = a(n-1)*5*(5*n - 1)*(5*n - 2)*(5*n - 3)*(5*n - 4)/n^4. - _Neven Sajko_, Jul 21 2023
%t Table[(5 n)!/(n)!^5, {n, 0, 20}] (* _Vincenzo Librandi_, Mar 08 2014 *)
%o (Magma) [Factorial(5*n)/Factorial(n)^5: n in [0..10]]; // _Vincenzo Librandi_, Mar 08 2014
%o (PARI) a(n) = (5*n)!/(n!)^5; \\ _Michel Marcus_, Mar 08 2014
%Y Cf. A000984, A006480, A008977, A002894, A002897, A186420, A188662, A001460.
%Y Row 5 of A187783.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_.
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