A174532 Expansion of x^5/((1-x)*(1+x-x^5)).

0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 2, -1, 3, -1, 2, 1, -1, 5, -5, 8, -6, 6, 0, -4, 13, -18, 25, -24, 21, -7, -10, 36, -59, 81, -87, 78, -41, -17, 99, -185, 264, -304, 288, -188, 4, 261, -564, 853, -1040, 1045, -783, 220, 634, -1673, 2719

Offset: 0

Author

Roger L. Bagula, Nov 28 2010

Keywords

• sign
• easy
• less

Crossrefs

Cf. A174522.

Programs

Magma

M:=Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1], [-1, 1, 0, 0, 1, 0]]); V:=Matrix([[0],[0],[0],[0],[0],[1]]); [ (M^n*V)[1][1]: n in [0..60] ]; // Klaus Brockhaus, Nov 30 2010

Mathematica

M = {{0, 1, 0, 0, 0, 0},
      {0, 0, 1, 0, 0, 0},
      {0, 0, 0, 1, 0, 0},
      {0, 0, 0, 0, 1, 0},
      {0, 0, 0, 0, 0, 1},
      {-1, 1, 0, 0, 1, 0}};
v[0] = Table[If[n == 6, 1, 0], {n, 1, 6}]
v[n_] := v[n] = M.v[n - 1];
b = Table[v[n][[1]], {n, 0, 50}]
CoefficientList[Series[x^5/((1-x)(1+x-x^5)),{x,0,60}],x] (* Harvey P. Dale, May 15 2021 *)

Formula

a(n) = first component of v(n) = M^n v(0), where v(0)=[0,0,0,0,0,1] and

M = {{0, 1, 0, 0, 0, 0},

{0, 0, 1, 0, 0, 0},

{0, 0, 0, 1, 0, 0},

{0, 0, 0, 0, 1, 0},

{0, 0, 0, 0, 0, 1},

{-1, 1, 0, 0, 1, 0}}.