Riordan matrices in the reciprocation of quadratic polynomials

We iterate contractive one-degree polynomials with coefficients in the ring K[[x]] of formal power series to calculate the reciprocal in K[[x]] of a quadratic polynomial. Doing this we meet the structure of Riordan array. We interpret certain changes of variable as a Riordan array. We finish the paper by using our techniques to find new ways to get known formulas for the sum of powers of natural numbers involving Stirling and Eulerian numbers.