The coefficient of x^10 in the power series expansion of (x^1 + x^2 + x^3 + ...)^2 is ___.

GATE 2016 · Discrete Mathematics · Generating Functions · medium

Answer: 9

1. Rewrite the series in closed form: x + x^2 + x^3 + ... = x(1 + x + x^2 + ...) = x/(1-x). Squaring: (x/(1-x))^2 = x^2/(1-x)^2.
2. Expand 1/(1-x)^2 as a power series: With k = 2: 1/(1-x)^2 = sum_{n>=0} C(n+1, 1) x^n = sum_{n>=0} (n+1) x^n. So x^2/(1-x)^2 = sum_{n>=0} (n+1) x^{n+2}.
3. Extract the coefficient of x^10: [x^10] x^2/(1-x)^2 = [x^8] 1/(1-x)^2 = 8 + 1 = 9.