Source

FractalTriangles.java - JAVA Applet

List of numbers

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 - First Kind of Stirling Numbers
3. f(n,k) = f(n-1,k-1) + k*f(n-1,k) mod M - Second Kind of Stirling Numbers
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) - First Kind of Euler Numbers
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) - Second Kind of Euler Numbers
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
    I found error here, see left triangle.
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

For each of numbers we assumed:

f(n,n) = 1
f(n,k) = 0 for n<k
f(n,0) = δ_n,0
f(0,k) = δ_0,k