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nC2 for n=0,1,2,3...
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Series will be: 0, 0, 1, 3, 6, 10, 15, 21, ....
Encoded in power series$: 1x^2 + 3x^3 + 6x^4 + 10x^5 + 15x^6 + 21x^7 + ...$
let,

$\ \ \ f(x) \ \ \ = 1x^2 + 3x^3 + 6x^4 + 10x^5 + 15x^6 + 21x^7 + ...$
$x*f(x) =           1x^3 + 3x^4 + 6x^5 + 10x^6 + 15x^7 + 21x^8 + ...$

 

Subtract the equations, we get:

$(1 - x)*f(x) = x^2 + 2x^3 + 3x^4 + 4x^5 + 5x^6 + 6x^7 + ...$
$(1 - x)*f(x) = x^2 (1 + 2x + 3x^2 + 4x^3 + 5x^4 + 6x^5 + ... )$

$(1 - x)*f(x) = \frac{x^2}{(1 - x)^2}$  (Generating function for $1 + 2x + 3x^2 + ...$ is $1/(1-x)^2$)

$f(x) = \frac{x^2}{(1 - x)^3}$

So $\frac{x^2}{(1 - x)^3}$ is the closed form of the given series.
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