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While it may be tempting to try, establishing plants from seeds in your new garden is very difficult.
First, a good rain will likely wash seeds away. Second, your rain garden needs deep plant roots to work. By using plants, you are "buying time" and will be that much closer to helping reduce runoff.
To calculate the number of plants required, for an 18" spacing, you can divide your rain garden's area in square feet by 2.25. Shrubs and trees can be used at greater spacing in large rain gardens.