terms (k,i): [(0, 1), (1, 1), (1, 2), (3, 1), (5, 1), (5, 2)]
c(0, 1)(p) = p**2*(28*p**3 - 70*p**2 - 94*p + 211)/75
c(1, 1)(p) = -p*(p - 1)*(2*p - 1)*(p**2 - p - 4)/4
c(1, 2)(p) = 9*p**2*(p - 1)**2*(2*p - 1)/16
c(3, 1)(p) = -7*p**2*(p - 1)**2*(2*p - 1)/24
c(5, 1)(p) = 71*p**2*(p - 1)**2*(2*p - 1)/200
c(5, 2)(p) = 7*p**2*(p - 1)**2*(2*p - 1)/80
check f(0) - p == 0: True
at p = 61/100:
  c(0, 1) = 41541389119/62500000000
  c(1, 1) = -1109016051/20000000000
  c(1, 2) = 560304459/80000000000
  c(3, 1) = -145264119/40000000000
  c(5, 1) = 4420179621/1000000000000
  c(5, 2) = 435792357/400000000000
c(1,2)(p) factored: 9*p**2*(p - 1)**2*(2*p - 1)/16
real roots of numerator of c(1,2): [0, 0, 0.500000000000, 1.00000000000, 1.00000000000]
  sign of c(1,2) on (0.50000000, 1.0000000): 1
