User:Georg Fischer/Generating functions and linear recurrences

A257890:

G.f.:          (x^2-x+1)*(x^2-3*x+3)/(x-1)^6.
Denominator:   (x-1)^6 = 1 - 6x + 15x^2 - 20x^3 + 15x^4 - 6x^5 + x^6
Nominator:     3 - 6x + 7x^2 - 4x^3 + x^4
l.r.signature: (6,-15,20,-15,6,-1).
Mathematica:   CoefficientList[Series[(x^2-x+1)*(x^2-3*x+3)/(x-1)^6, {x,0,14}], x]
               LinearRecurrence[{6, -15, 20, -15, 6, -1}, {3, 12, 34, 80, 166, 314}, 14]
PARI:          Vec((x^2-x+1)*(x^2-3*x+3)/(x-1)^6 + O(x^14))
DATA:          3, 12, 34, 80, 166, 314, 553, 920, 1461, 2232, 3300, 4744, 6656, 9142

A295333 (Robert Israel, Nov 28 2017):

f:= gfun:-rectoproc({a(n)+6*a(n+3)-a(n+6), a(0) = 1, a(1) = 2, a(2) = 3, a(3) = 8, a(4) = 11, a(5) = 19},a(n),remember):
G.f.: G(x) = (1 + 2*x + 3*x^2 + 2*x^3 - x^4 + x^5)/(1 - 6*x^3 - x^6).