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A071549 a(n) = (7n)!/n!^7.

%I A071549 #27 Feb 06 2024 08:07:19
%S A071549 1,5040,681080400,182509367040000,66475579247327250000,
%T A071549 28837919555681211870935040,14007180988362844601443040716800,
%U A071549 7363615666157189603982585462030336000,4104167472585675600759440022842715359250000,2392741010223442438553822446842770682716580000000
%N A071549 a(n) = (7n)!/n!^7.
%C A071549 Number of closed paths of length 7n whose steps are 7th roots of unity. - _Andrew Howroyd_, Nov 01 2018
%H A071549 Vincenzo Librandi, <a href="/A071549/b071549.txt">Table of n, a(n) for n = 0..100</a>
%H A071549 Gilbert Labelle and Annie Lacasse, <a href="https://doi.org/10.46298/dmtcs.2937">Closed paths whose steps are roots of unity</a>, in FPSAC 2011, Reykjavik, Iceland DMTCS proc. AO, 2011, 599-610.
%H A071549 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 A071549 From _Peter Bala_, Feb 14 2020: (Start)
%F A071549 a(n) = C(7*n,n)*C(6*n,n)*C(5*n,n)*C(4*n,n)*C(3*n,n)*C(2*n,n).
%F A071549 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 A071549 a(n) = [x^n](F(x)^(5040*n)), where F(x) = 1 + x + 62528*x^2 + 11087269661*x^3 + 3021437267047869*x^4 + 1045823730475703710735*x^5 + ...
%F A071549 appears to have integer coefficients. For similar results see A008979.
%F A071549 a(n) = [(x*y*z*u*v*w)^n] (1 + x + y + z + u + v + w)^(7*n). (End)
%t A071549 Table[(7 n)!/(n)!^7, {n, 0, 20}] (* _Vincenzo Librandi_, Aug 13 2014 *)
%o A071549 (Magma) [Factorial(7*n)/Factorial(n)^7: n in [0..20]]; // _Vincenzo Librandi_, Aug 13 2014
%o A071549 (PARI) a(n) = (7*n)!/(n!^7); \\ _Andrew Howroyd_, Nov 01 2018
%Y A071549 Row n=7 of A187783, column k=7 of A089759.
%Y A071549 Sequences (k*n)!/n!^k: A000984 (k = 2), A006480 (k =3), A008977 (k = 4), A008978 (k = 5), A008979 (k = 6), A071550 (k = 8), A071551 (k = 9), A071552 (k = 10).
%K A071549 nonn
%O A071549 0,2
%A A071549 _Benoit Cloitre_, May 30 2002
%E A071549 a(8)-a(9) added by _Andrew Howroyd_, Nov 01 2018