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Consider the special case when you multiply a sequence by \(1, 1, 1, \ldots\text{.}\) For example, multiply \(1,1,1,\ldots\) by \(1, 2, 3, 4, 5\ldots\text{.}\) The first term is \(1\cdot 1 = 1\text{.}\) Then \(1\cdot 2 + 1 \cdot 1 = 3\text{.}\) Then \(1\cdot 3 + 1\cdot 2 + 1 \cdot 1 = 6\text{.}\) The next term will be 10. We are getting the triangular numbers. More precisely, we get the sequence of partial sums of \(1,2,3,4,5, \ldots\text{.}\) In terms of generating functions, we take \(\frac{1}{1-x}\) (generating \(1,1,1,1,1\ldots\)) and multiply it by \(\frac{1}{(1-x)^2}\) (generating \(1,2,3,4,5,\ldots\)) and this give \(\frac{1}{(1-x)^3}\text{.}\) This should not be a surprise as we found the same generating function for the triangular numbers earlier.