Consider the polynomial sequence p_0 = x^0 p_1 = x^1 p_2 = x^2 + 1x^1 p_3 = x^3 + 3x^2 + 1x^1 p_4 = x^4 + 6x^3 + 7x^2 + 1x^1 p_5 = x^5 + 10x^4 + 25x^3 + 15x^2 + 1x^1 ... where the coefficients here are the Stirling numbers of the second kind. Then {p_n} is of binomial type.
I'm puzzled why this fact isn't quite so famous as I'd say it should be. (At any rate, I've never before seen it mentioned.) It would seem, for example, to provide an alternate definition of the Stirling numbers of the second kind -- which are glamorous combinatorial celebrities.