The new constant term is just \(1 \cdot 1\text{.}\) The next term will be \(1\cdot 2 + 2 \cdot 1 = 4\text{.}\) The next term: \(1 \cdot 4 + 2 \cdot 2 + 3 \cdot 1 = 11\text{.}\) One more: \(1 \cdot 8 + 2 \cdot 4 + 3 \cdot 2 + 4 \cdot 1 = 26\text{.}\) The resulting sequence is

Since the generating function for \(1,2,3,4, \ldots\) is \(\frac{1}{(1-x)^2}\) and the generating function for \(1,2,4,8, 16, \ldots\) is \(\frac{1}{1-2x}\text{,}\) we have that the generating function for \(1,4, 11, 26, 57, \ldots\) is \(\frac{1}{(1-x)^2(1-2x)}\)