Fractal face of Pascal Triangle

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Source: FractalTriangles.java

List of recurrences:

  1. f(n,k) = f(n-1,k-1) + f(n-1,k) mod M - Pascal Triangle
  2. f(n,k) = f(n-1,k-1) + (n-1)*f(n-1,k) mod M - Stirling Numbers of the first kind
  3. f(n,k) = f(n-1,k-1) + k*f(n-1,k) mod M - Stirling Numbers of the second kind
  4. f(n,k) = (n-k)*f(n-1,k-1) + (k+1)*f(n-1,k) mod M, (for n==k f(n,k)=0) - Euler Numbers of the first kind
  5. f(n,k) = (2*n-k-1)*f(n-1,k-1) + (k+1)*f(n-1,k) mod M, (for n==k f(n,k)=0) - Euler Numbers of the second kind
  6. f(n,k) = f(n-1,k-1) + f(n-1,k)*f(n-1,k) mod M
  7. f(n,k) = f(n-1,k-1)*f(n-1,k-1) + f(n-1,k) mod M
  8. f(n,k) = f(n-1,k-1)*f(n-1,k-1) + f(n-1,k)*f(n-1,k) mod M
  9. f(n,k) = (n-1)*f(n-1,k-1) + (k+1)*f(n-1,k) mod M
  10. f(n,k) = (n+1)*f(n-1,k-1) + (k-1)*f(n-1,k) mod M
  11. f(n,k) = f[n-k-1)*f(n-1,k-1) + f[k+1)*f(n-1,k) mod M
  12. f(n,k) = (f[n-k+1)*f(n-1,k-1) + f[k-1)*f(n-1,k)) mod M
  13. f(n,k) = f(n-1,k-1) + f[k)*f(n-1,k) mod M
  14. f(n,k) = f(n-1,k-1) + f[n)*f(n-1,k) mod M
  15. f(n,k) = f(n-1,k+1) + f(n-1,k) + f(n-1,k-1) mod M
  16. f(n,k) = f(n-1,k+1) + f(n-1,k-1) mod M
  17. for (n > 1) f(n,k) = f(n-1,k+1) + f(n-2,k) + f(n-1,k-1) mod M
    else f(n,k) = f(n-1,k+1) + mod + f(n-1,k-1) mod M
  18. f(n,k) = f(n-1,k-1)^f(n-1,k) + f(n-1,k)^f(n-1,k-1) mod M
  19. f(n,k) = f(n-1,k-1) + f(n-1,k)^f(n-1,k-1) mod M
  20. f(n,k) = f(n-1,k) + f(n-1,k)^f(n-1,k-1) mod M
  21. f(n,k) = f(n-1,k) + (f(n-1,k-1)^f(n-1,k) mod k) mod M
  22. f(n,k) = f(n-1,k) + (f(n-1,k-1)^f(n-1,k) mod n) mod M
  23. f(n,k) = f(n-1,k) + (f(n-1,k-1)^f(n-1,k) mod M-f(n-1,k-1)) mod M
  24. f(n,k) = f(n-1,k) + (f(n-1,k-1)^f(n-1,k) mod M-f(n-1,k)) mod M

Initial conditions:

f(n,n) = 1, f(n,k) = 0 for n<k, f(n,0) = δn,0, f(0,k) = δ0,k.