A290535 Determinant of circulant matrix of order 6 with entries in the first row (-1)^(j-1)*Sum_{k>=0} (-1)^k*binomial(n, 6*k + j - 1), j=1..6.

1, 0, 0, 0, 0, 0, -104284, -783050688, -329029322076, -43271152876224, -2175830808446736, 0, 5427970251634650916, 307609249050423946080, 8866068073884849492756, 137518739026000524646272, 896278292839676023110288, 0, -2518571790589921864549097500

Offset: 0

Author

Vladimir Shevelev and Peter J. C. Moses, Aug 05 2017

Keywords

• sign

Comments

a(n) = 0 for n == 5 (mod 6).

Links

• Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
• Wikipedia, Circulant matrix

Crossrefs

Cf. A290285, A290286.

Programs

Mathematica

ro[n_] := Table[(-1)^(j-1) Sum[(-1)^k Binomial[n, 6k+j-1], {k, 0, n/6}], {j, 1, 6}];
M[n_] := Table[RotateRight[ro[n], m], {m, 0, 5}];
a[n_] := Det[M[n]];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Aug 10 2018 *)