Question

Find the generating function of the series $3+5x+9x^2+15x^3+23x^4+33x^5+....$

Answer 1

Forming the successive orders of differences of the coefficients, we have the series
$2,4,6,8,10,....$
$2,2,2,2,.....;$
Thus the terms in the second order of differences are equal$;$ hence $a_n$ is a rational integral function of n of two dimensions$;$ and therefore the scale of relation is $(1-x{)}^3$. We have
$S=3+5x+9x^2+15x^3+23x^4+33x^5+...$
$-3xS=-9x-15x^2-27x^3-45x^4-69x^5-....$
$3x^{2}S=9x^2+15x^3+27x^4+45x^5+....$
$-x^{3}S=-3x^3-5x^4-9x^5-....$
By adding$,$ $(1-x{)}^3.S=3-4x+3x^2;$
$\therefore S=\frac{3-4x+3x^2}{(1-x{)}^3}$.