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

1, 5040, 681080400, 182509367040000, 66475579247327250000, 28837919555681211870935040, 14007180988362844601443040716800, 7363615666157189603982585462030336000, 4104167472585675600759440022842715359250000, 2392741010223442438553822446842770682716580000000

Offset: 0

Author

Benoit Cloitre, May 30 2002

Keywords

• nonn

Comments

Number of closed paths of length 7n whose steps are 7th roots of unity. - Andrew Howroyd, Nov 01 2018

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100
• Gilbert Labelle and Annie Lacasse, Closed paths whose steps are roots of unity, in FPSAC 2011, Reykjavik, Iceland DMTCS proc. AO, 2011, 599-610.
• R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.

Crossrefs

Row n=7 of A187783, column k=7 of A089759.

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).

Programs

Magma

[Factorial(7*n)/Factorial(n)^7: n in [0..20]]; // Vincenzo Librandi, Aug 13 2014

Mathematica

Table[(7 n)!/(n)!^7, {n, 0, 20}] (* Vincenzo Librandi, Aug 13 2014 *)

PARI

a(n) = (7*n)!/(n!^7); \\ Andrew Howroyd, Nov 01 2018

Formula

From Peter Bala, Feb 14 2020: (Start)

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).

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.

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 + ...

appears to have integer coefficients. For similar results see A008979.

a(n) = [(x*y*z*u*v*w)^n] (1 + x + y + z + u + v + w)^(7*n). (End)

Extensions

a(8)-a(9) added by Andrew Howroyd, Nov 01 2018