In reply to elon.

Look at the first graph in this article and re-read the section "Output and visualize spline effects." The graph shows that the spline effects consist of an intercept, a linear term, and (restricted) cubic polynomials. The Spl_1 term is linear. The Spl_2 and Spl_3 terms are cubic. Thus, the model is the same as you present. The article's formula for the predicted values is correct. The predicted values are linear combinations of the effects in the model.