Part 1: identity r_n = p(1^n) Delta_n / (rho^2 C(n,2)^2 C(n-2,2)) for random rate arrays, N = 7
  trial 0 rho=3 n=4 : residual == prediction : True   (Delta_n = 336549)
  trial 0 rho=3 n=5 : residual == prediction : True   (Delta_n = 1.18771e+06)
  trial 0 rho=3 n=6 : residual == prediction : True   (Delta_n = 133060)
  trial 0 rho=3 n=7 : residual == prediction : True   (Delta_n = 36247.1)
  trial 1 rho=5/2 n=4 : residual == prediction : True   (Delta_n = 34713)
  trial 1 rho=5/2 n=5 : residual == prediction : True   (Delta_n = 101268)
  trial 1 rho=5/2 n=6 : residual == prediction : True   (Delta_n = 225367)
  trial 1 rho=5/2 n=7 : residual == prediction : True   (Delta_n = 64519)
  trial 2 rho=7/3 n=4 : residual == prediction : True   (Delta_n = 45955.1)
  trial 2 rho=7/3 n=5 : residual == prediction : True   (Delta_n = 267652)
  trial 2 rho=7/3 n=6 : residual == prediction : True   (Delta_n = 391064)
  trial 2 rho=7/3 n=7 : residual == prediction : True   (Delta_n = 136989)
  trial 3 rho=2 n=4 : residual == prediction : True   (Delta_n = 106754)
  trial 3 rho=2 n=5 : residual == prediction : True   (Delta_n = 350851)
  trial 3 rho=2 n=6 : residual == prediction : True   (Delta_n = 161326)
  trial 3 rho=2 n=7 : residual == prediction : True   (Delta_n = 172378)
Part 1 all identities exact: True

Part 2: paintbox closed forms vs Schweinsberg type sums (m <= 8)
  a=1/3 atoms=[('1', ['1/4', '1/8'])] : all m<=8 agree: True
  a=0 atoms=[('2', ['1/3', '1/5', '1/7']), ('1/2', ['1/2'])] : all m<=8 agree: True
  a=1/5 atoms=[('1', ['1/2', '1/2']), ('3', ['1/6', '1/6', '1/6'])] : all m<=8 agree: True
Part 2 all agree: True

Part 3: Delta_n for Xi = (17/1024) delta_0 + delta_(1/4,1/8), rho = 13833/5120
  n= 4  Delta_n = 0   X(n)Phi(n-1)/C(n,2) = 0.133813   Bin(n)W(n-1)/C(n,2) = 0.133813
  n= 5  Delta_n = 0   X(n)Phi(n-1)/C(n,2) = 0.356039   Bin(n)W(n-1)/C(n,2) = 0.356039
  n= 6  Delta_n = 0.273637 (sign +1)   X(n)Phi(n-1)/C(n,2) = 0.575357   Bin(n)W(n-1)/C(n,2) = 0.557115
  n= 7  Delta_n = 1.19188 (sign +1)   X(n)Phi(n-1)/C(n,2) = 0.72995   Bin(n)W(n-1)/C(n,2) = 0.673194
  n= 8  Delta_n = 2.84873 (sign +1)   X(n)Phi(n-1)/C(n,2) = 0.800492   Bin(n)W(n-1)/C(n,2) = 0.698751
  n= 9  Delta_n = 4.92959 (sign +1)   X(n)Phi(n-1)/C(n,2) = 0.796461   Bin(n)W(n-1)/C(n,2) = 0.659528
  n=10  Delta_n = 6.85795 (sign +1)   X(n)Phi(n-1)/C(n,2) = 0.739513   Bin(n)W(n-1)/C(n,2) = 0.587115
  n=11  Delta_n = 8.03792 (sign +1)   X(n)Phi(n-1)/C(n,2) = 0.652306   Bin(n)W(n-1)/C(n,2) = 0.506162
  n=12  Delta_n = 8.0379 (sign +1)   X(n)Phi(n-1)/C(n,2) = 0.553195   Bin(n)W(n-1)/C(n,2) = 0.431408
  n=16  Delta_n = -4.8126 (sign -1)   X(n)Phi(n-1)/C(n,2) = 0.221345   Bin(n)W(n-1)/C(n,2) = 0.26145
  n=20  Delta_n = -28.7172 (sign -1)   X(n)Phi(n-1)/C(n,2) = 0.0693649   Bin(n)W(n-1)/C(n,2) = 0.220508
  n=25  Delta_n = -59.8936 (sign -1)   X(n)Phi(n-1)/C(n,2) = 0.0134419   Bin(n)W(n-1)/C(n,2) = 0.213087
  n=30  Delta_n = -91.4514 (sign -1)   X(n)Phi(n-1)/C(n,2) = 0.00228045   Bin(n)W(n-1)/C(n,2) = 0.212513
  n=40  Delta_n = -165.709 (sign -1)   X(n)Phi(n-1)/C(n,2) = 5.14112e-05   Bin(n)W(n-1)/C(n,2) = 0.212499
real 0.08
user 0.07
sys 0.00
