Hi,
these coefficients occur exactly in the powers of the matrix of Stirling numbers 2'nd kind, as I described it in my posting about iteration for e^x-1 and which must be used in a factorial scaled version to obtain the results.
Your coefficients in rows for a certain iteration are the coefficients of the second column of the related power of St2 (second column only will be used for the scalar result in my notation).
second iteration = second power
third iteration = third power
P.s. If you need a free numb-theoretical package with arbitrary precision you might consider Pari/GP I've also written a convenient user-interface for this usable in win-xp ("Pari-TTY")
Gottfried
these coefficients occur exactly in the powers of the matrix of Stirling numbers 2'nd kind, as I described it in my posting about iteration for e^x-1 and which must be used in a factorial scaled version to obtain the results.
Your coefficients in rows for a certain iteration are the coefficients of the second column of the related power of St2 (second column only will be used for the scalar result in my notation).
Code:
VE(St2,8,6): \\ extraction of 8 rows, 6 columns
1 . . . . . ...
0 1 . . . .
0 1 1 . . .
0 1 3 1 . .
0 1 7 6 1 .
0 1 15 25 10 1
0 1 31 90 65 15
0 1 63 301 350 140 ...
... ...
second iteration = second power
Code:
VE(St2^2,8,6):
1 . . . . . ...
0 1 . . . .
0 2 1 . . .
0 5 6 1 . .
0 15 32 12 1 .
0 52 175 110 20 1
0 203 1012 945 280 30
0 877 6230 8092 3465 595 ...
... ...
third iteration = third power
Code:
VE(St2^3,8,6):
1 . . . . .
0 1 . . . .
0 3 1 . . .
0 12 9 1 . .
0 60 75 18 1 .
0 358 660 255 30 1
0 2471 6288 3465 645 45
0 19302 65051 47838 12495 1365
... ...
P.s. If you need a free numb-theoretical package with arbitrary precision you might consider Pari/GP I've also written a convenient user-interface for this usable in win-xp ("Pari-TTY")
Gottfried
Gottfried Helms, Kassel