Given that n is a whole number, what is the next expression in this series of expressions, each of which is valid for all n?

n^0,
2n+1,
8n^2+6n+1,
32n^3+40n^2+12n+1

(In reply to re: One possible answer to what's next by broll)

   0 _1_ ___2___ ___3___ ___4___ __5___ ...
1) 1 0×1 __0×2^3 __0×2^5 __0×2^7 _0×2^9 ...
2) 1 1×2 __0×2^3 __0×2^5 __0×2^7 _0×2^9 ...
3) 1 2×3 __1×2^3 __0×2^5 __0×2^7 _0×2^9 ...
4) 1 3×4 __5×2^3 __1×2^5 __0×2^7 _0×2^9 ...
5) 1 4×5 _15×2^3 __7×2^5 __1×2^7 _0×2^9 ...
6) 1 5×6 _35×2^3 _28×2^5 __9×2^7 _1×2^9 ...
7) 1 6×7 _70×2^3 _84×2^5 _45×2^7 11×2^9 ...
8) ...

0) 1
1) n(n+1)
2) binomial coefficients C(n, 4) ×2^ 3
3) binomial coefficients C(n, 6) ×2^ 5
4) binomial coefficients C(n, 8) ×2^ 7
5) binomial coefficients C(n,10) ×2^ 9
6) binomial coefficients C(n,12) ×2^11
...
N) binomial coefficients C(n,2N) ×2^(2N-1)

[The nth term for each sequence of binomial coefficient C(n, x) as it relates to the polynomial begins with 1 for the Nth coefficient of each  polynomial].

Edited on June 27, 2011, 1:16 pm

Posted by Dej Mar on 2011-06-27 12:55:42