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How to find pattern with Mathematica?

• To: mathgroup at smc.vnet.net
• Subject: [mg104029] How to find pattern with Mathematica?
• From: barefoot gigantor <barefoot1980 at gmail.com>
• Date: Fri, 16 Oct 2009 07:19:16 -0400 (EDT)

What is the pattern in the following polynom: (how to generate them)

1:     12 n^2 + 12 n + 1

2:    300 n^3 + 450 n^2 + 160 n + 5

3:   840 n^4+1680 n^3+1030 n^2+190 n+3

4:  1260 n^5+3150 n^4+2730 n^3+945 n^2+107 n+1

5:  27720 n^6+83160 n^5+93030 n^4+47460 n^3+10689 n^2+819 n+5