Question

generating function for 0,2,6,12,20,30,42,....

Answer 1

The formula is n×(n-1)
here,
1×0=0
2×1=2
3×2=6
4×3=12
5×4=20
6×5=30
7×6=42
8×7=56.

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Answer 2

"56, 72, 90,....and so on".

Explanation :    

In discrete mathematics, a generating function is a method used for encoding an endless sequence of numbers ($a_{n}$) by treating them as the coefficients of a proper power series.

This is an extremely powerful tool in discrete mathematics used to manipulate sequences. Generating function is also a method used to solve the recurrence relations.

Here we use, $a_{n} = n^{2} + n$

$a_{0} = 0^{2} + 0 = 0$

$a_{1} = 1^{2} + 1 = 2$

$a_{2} = 2^{2} + 2 = 4+2 = 6$

$a_{3} = 3^{2} + 3 = 9 + 3 = 12$

$a_{4} = 4^{4} + 4 = 16 + 4 = 20$

$a_{5} = 5^{2} + 5 = 25 + 5 = 30$

$a_{6} = 6^{2} + 6 = 36 + 6 = 42$

$a_{7} = 7^{2} + 7 = 49 + 7 = 56$

$a_{8} = 8^{2} + 8 = 64 + 8 = 72$

$a_{9} = 9^{2} + 9 = 81 + 9 = 90$

and so on.