Putnam 1949 B4

Show that the coefficients $a_1 , a_2 , a_3 ,\ldots$ in the expansion $$\frac{1}{4}\left(1+x-\frac{1}{\sqrt{1-6x+x^{2}}}\right) =a_{1} x+ a_2 x^2 + a_3 x^3 +\ldots$$ are positive integers.